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In this paper, various tensorial inequalities of trapezoid type were obtained. Identity from classical analysis is utilized to obtain the tensorial version of the said identity which in turn allowed us to obtain tensorial inequalities in Hilbert space. The continuous functions of self-adjoint operators in Hilbert spaces have several tensorial norm inequalities discovered in this study. The convexity features of the mapping f lead to the variation in several inequalities of the trapezoid type. [ABSTRACT FROM AUTHOR]

Starlike and convex functions have gained increased prominence in both academic literature and practical applications over the past decade. Concurrently, logarithmic coefficients play a pivotal role in estimating diverse properties within the realm of analytic functions, whether they are univalent or nonunivalent. In this paper, we rigorously derive bounds for specific Toeplitz determinants involving logarithmic coefficients pertaining to classes of convex and starlike functions concerning symmetric points. Furthermore, we present illustrative examples showcasing the sharpness of these established bounds. Our findings represent a substantial contribution to the advancement of our understanding of logarithmic coefficients and their profound implications across diverse mathematical contexts. [ABSTRACT FROM AUTHOR]

In this paper, we present two new classes of fuzzy negations. They are an extension of a well-known class of fuzzy negations, the Sugeno Class. We use it as a base for our work for the first two construction methods. The first method generates rational fuzzy negations, where we use a second-degree polynomial with two parameters. We investigate which of these two conditions must be satisfied to be a fuzzy negation. In the second method, we use an increasing function instead of the parameter δ of the Sugeno class. In this method, using an arbitrary increasing function with specific conditions, fuzzy negations are produced, not just rational ones. Moreover, we compare the equilibrium points of the produced fuzzy negation of the first method and the Sugeno class. We use the equilibrium point to present a novel method which produces strong fuzzy negations by using two decreasing functions which satisfy specific conditions. We also investigate the convexity of the new fuzzy negation. We give some conditions that coefficients of fuzzy negation of the first method must satisfy in order to be convex. We present some examples of the new fuzzy negations, and we use them to generate new non-symmetric fuzzy implications by using well-known production methods of non-symmetric fuzzy implications. We use convex fuzzy negations as decreasing functions to construct an Archimedean copula. Finally, we investigate the quadratic form of the copula and the conditions that the coefficients of the first method and the increasing function of the second method must satisfy in order to generate new copulas of this form. [ABSTRACT FROM AUTHOR]

Hermite–Hadamard inequalities and their refinements have been investigated for a long period of time. In this paper, we obtained refinements of the Hermite–Hadamard inequality of tensorial type for the convex functions of self-adjoint operators in Hilbert spaces. The obtained inequalities generalize the previously obtained inequalities by Dragomir. We also provide useful Lemmas which enabled us to obtain the results. The examples of the obtained inequalities for specific convex functions have been given in the example and consequences section. Symmetry in the upper and lower bounds can be seen in the last Theorem of the paper given, as the upper and lower bounds differ by a constant. [ABSTRACT FROM AUTHOR]

In this paper, a weakly coupled system (by the displacement of symmetric type) consisting of a viscoelastic Kirchhoff plate equation involving free boundary conditions and the viscoelastic wave equation with Dirichlet boundary conditions in a bounded domain is considered. Under the assumptions on a more general type of relaxation functions, an explicit and general decay rate result is established by using the multiplier method and some properties of the convex functions. [ABSTRACT FROM AUTHOR]

The purpose of this paper was to define a new class of close-to-convex function, denoted by C V (δ , α) , which is a subclass of all functions that are univalent in D and have positive coefficients normalized by the conditions f (0) = 0 ,     f ′ (0) = 1 , if it satisfies such a condition that is dependent on positive real part. Furthermore, we proved how the power series distribution is essential for determining the sufficient and necessary condition on any function f in class C V (δ , α) . [ABSTRACT FROM AUTHOR]

In this paper, new Fejér-type inequalities for harmonically convex functions are obtained. Some mappings related to the Fejér-type inequalities for harmonically convex are defined. Properties of these mappings are discussed and, as a consequence, we obtain refinements of some known results. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to present classical second-order differential subordination knowledge extended in this study to include new results regarding third-order differential subordinations. The focus of this study is on the main problems examined by differential subordination theory. Hence, the new results obtained here reveal techniques for identifying dominants and the best dominant of certain third-order differential subordinations. Another aspect of novelty is the new application of the Gaussian hypergeometric function. Novel third-order differential subordination results are obtained using the best dominant provided by the theorems and the operator previously defined as Gaussian hypergeometric function's fractional integral. The research investigation is concluded by giving an example of how the results can be implemented. [ABSTRACT FROM AUTHOR]

The q-symmetric analogues of Hölder, Minkowski, and power mean inequalities are presented in this paper. The obtained inequalities along with a Montgomery identity involving q-symmetric integrals are used to extend some Ostrowski-type inequalities. The q-symmetric derivatives of the functions involved in these Ostrowski-type inequalities are convex or s-convex. Moreover, some Hermite–Hadamard inequalities for convex functions as well as for s-convex functions are also acquired with the help of q-symmetric calculus in the present work. Some examples are included to support the effectiveness of the proved results. [ABSTRACT FROM AUTHOR]

In this investigation, we first establish new quantum Hermite–Hadamard type integral inequalities for s-convex functions by utilizing newly defined T q -integrals. Then, by using obtained inequality, we establish a new Hermite–Hadamard inequality for coordinated s 1 , s 2 -convex functions. The results obtained in this paper provide significant extensions of other related results given in the literature. Finally, some examples are given to illustrate the result obtained in this paper. These types of analytical inequalities, as well as solutions, apply to different areas where the concept of symmetry is important. [ABSTRACT FROM AUTHOR]

The relevance of convex and non-convex functions in optimization research is well known. Due to the behavior of its definition, the idea of convexity also plays a major role in the subject of inequalities. The main concern of this paper is to establish new integral inequalities for newly defined left and right convex interval-valued function on coordinates through pseudo order relation and double integral. Some of the Hermite–Hadamard type inequalities for the product of two left and right convex interval-valued functions on coordinates are also obtained. Moreover, Hermite–Hadamard–Fejér type inequalities are also derived for left and right convex interval-valued functions on coordinates. Some useful examples are also presented to prove the validity of this study. The proved results of this paper are generalizations of many known results, which are proved by Dragomir, Latif et al. and Zhao, and can be vied as applications of this study. [ABSTRACT FROM AUTHOR]

There is significant interaction between the class of symmetric functions and other types of functions. The multiplicative convex function class, which is intimately related to the idea of symmetry, is one of them. In this paper, we obtain some new generalized multiplicative fractional Hermite–Hadamard type inequalities for multiplicative convex functions and for their product. Additionally, we derive a number of inequalities for multiplicative convex functions related to generalized multiplicative fractional integrals utilising a novel identity as an auxiliary result. We provide some examples for the appropriate selections of multiplicative convex functions and their graphical representations to verify the authenticity of our main results. [ABSTRACT FROM AUTHOR]

The class of symmetric function interacts extensively with other types of functions. One of these is the class of convex functions, which is closely related to the theory of symmetry. In this paper, we obtain some new fractional Hermite–Hadamard inequalities with an exponential kernel for subadditive functions and for their product, and some known results are recaptured. Moreover, using a new identity as an auxiliary result, we deduce several inequalities for subadditive functions pertaining to the new fractional integrals involving an exponential kernel. To validate the accuracy of our results, we offer some examples for suitable choices of subadditive functions and their graphical representations. [ABSTRACT FROM AUTHOR]

This paper investigates several integral inequalities held simultaneously for q and h-integrals in implicit form. These inequalities are established for symmetric functions using certain types of convex functions. Under certain conditions, Hadamard-type inequalities are deducible for q-integrals. All the results are applicable for ℏ-convex, m-convex and convex functions defined on the non-negative part of the real line. [ABSTRACT FROM AUTHOR]

In the present paper, we first prove a new integral identity. Using that identity, we establish some fractional weighted midpoint-type inequalities for functions whose first derivatives are extended s-convex. Some special cases are discussed. Finally, to prove the effectiveness of our main results, we provide some applications to numerical integration as well as special means. [ABSTRACT FROM AUTHOR]

This paper is devoted to investigating the existence of solutions for the fractional differential equation and fractional differential inclusion of order α ∈ (2 , 3 ] with affine periodic boundary value conditions. Applying the Leray–Schauder fixed point theorem, the existence of the solutions for the fractional differential equation is established. Furthermore, for the fractional differential inclusion, we consider two cases: (i) the set-valued function has convex value and (ii) the set-valued function has nonconvex value. The main tools of our research are the Leray–Schauder alternative theorem, Covita and Nadler's fixed point theorem and some set-valued analysis theories. [ABSTRACT FROM AUTHOR]

This study deals with a novel class of mean-type inequalities by employing fractional calculus and convexity theory. The high correlation between symmetry and convexity increases its significance. In this paper, we first establish an identity that is crucial in investigating fractional mean inequalities. Then, we establish the main results involving the error estimation of the Hermite–Hadamard inequality for composite convex functions via a generalized Riemann-type fractional integral. Such results are verified by choosing certain composite functions. These results give well-known examples in special cases. The main consequences can generalize many known inequalities that exist in other studies. [ABSTRACT FROM AUTHOR]

Various scholars have lately employed a wide range of strategies to resolve specific types of symmetrical fractional differential equations. In this paper, we propose a new fractional identity for multiplicatively differentiable functions; based on this identity, we establish some new fractional multiplicative Bullen-type inequalities for multiplicative differentiable convex functions. Some applications of the obtained results are given. [ABSTRACT FROM AUTHOR]

In this article, we provide constraints for the sum by employing a generalized modified form of fractional integrals of Riemann-type via convex functions. The mean fractional inequalities for functions with convex absolute value derivatives are discovered. Hermite–Hadamard-type fractional inequalities for a symmetric convex function are explored. These results are achieved using a fresh and innovative methodology for the modified form of generalized fractional integrals. Some applications for the results explored in the paper are briefly reviewed. [ABSTRACT FROM AUTHOR]

In the development of algorithms for convex optimization problems, symmetry plays a very important role in the approximation of solutions in various real-world problems. In this paper, based on a fixed point algorithm with the inertial technique, we proposed and study a new accelerated algorithm for solving a convex bilevel optimization problem for which the inner level is the sum of smooth and nonsmooth convex functions and the outer level is a minimization of a smooth and strongly convex function over the set of solutions of the inner level. Then, we prove its strong convergence theorem under some conditions. As an application, we apply our proposed algorithm as a machine learning algorithm for solving some data classification problems. We also present some numerical experiments showing that our proposed algorithm has a better performance than the five other algorithms in the literature, namely BiG-SAM, iBiG-SAM, aiBiG-SAM, miBiG-SAM and amiBiG-SAM. [ABSTRACT FROM AUTHOR]

In this paper, we present some mappings defined over 0 , 1 related to the Fejér-type inequalities that have been established for harmonically convex functions. As a consequence, we obtain companions of Fejér-type inequalities for harmonically convex functions by using these mappings. Properties of these mappings are discussed, and consequently, we obtain refinement inequalities of some known results. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to establish some new variants of the Jensen–Mercer inequality via harmonically strongly convex function. We also propose some new fractional analogues of Hermite–Hadamard–Jensen–Mercer-like inequalities using AB fractional integrals. In order to obtain some of our main results, we also derive new fractional integral identities. To demonstrate the significance of our main results, we present some interesting applications to special means and to error bounds as well. [ABSTRACT FROM AUTHOR]

As is known, integral inequalities related to convexity have a close relationship with symmetry. In this paper, we introduce a new notion of interval-valued harmonically m , h 1 , h 2 -Godunova–Levin functions, and we establish some new Hermite–Hadamard inequalities. Moreover, we show how this new notion of interval-valued convexity has a close relationship with many existing definitions in the literature. As a result, our theory generalizes many published results. Several interesting examples are provided to illustrate our results. [ABSTRACT FROM AUTHOR]

The research presented in this paper deals with analytic p-valent functions related to the generalized probability distribution in the open unit disk U. Using the Hadamard product or convolution, function f s (z) is defined as involving an analytic p-valent function and generalized distribution expressed in terms of analytic p-valent functions. Neighborhood properties for functions f s (z) are established. Further, by applying a previously introduced linear transformation to f s (z) and using an extended Libera integral operator, a new generalized Libera-type operator is defined. Moreover, using the same linear transformation, subclasses of starlike, convex, close-to-convex and spiralike functions are defined and investigated in order to obtain geometrical properties that characterize the new generalized Libera-type operator. Symmetry properties are due to the involvement of the Libera integral operator and convolution transform. [ABSTRACT FROM AUTHOR]

Fractional integrals and inequalities have recently become quite popular and have been the prime consideration for many studies. The results of many different types of inequalities have been studied by launching innovative analytical techniques and applications. These Hermite–Hadamard inequalities are discovered in this study using Atangana–Baleanu integral operators, which provide both practical and powerful results. In this paper, a symmetric study of integral inequalities of Hermite–Hadamard type is provided based on an identity proved for Atangana–Baleanu integral operators and using functions whose absolute value of the second derivative is harmonic convex. The proven Hermite–Hadamard-type inequalities have been observed to be valid for a choice of any harmonic convex function with the help of examples. Moreover, fractional inequalities and their solutions are applied in many symmetrical domains. [ABSTRACT FROM AUTHOR]

Integral inequalities play critical roles inmeasure theory and probability theory. Given recent profound discoveries in the field of fuzzy set theory, fuzzy inequality has become a hot research topic in recent years. For classical Sandor type inequality, this paper intends to extend the Sugeno integral. Based on the (s,m)-convex function in the second sense, a new Sandor type inequality is proposed for the Sugeno integral. Examples are given to verify the conclusion of this paper. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to define a new family of Szász–Mirakyan operators that depends on a non-negative parameter, say α. This new family of Szász–Mirakyan operators is crucial in that it includes both the existing Szász–Mirakyan operator and allows the construction of new operators for different values of α. Then, the convergence properties of the new operators with the aid of the Popoviciu–Bohman–Korovkin theorem-type property are presented. The Voronovskaja-type theorem and rate of convergence are provided in a detailed proof. Furthermore, with the help of the classical modulus of continuity, we deduce an upper bound for the error of the new operator. In addition to these, in order to show that the convex or monotonic functions produced convex or monotonic operators, we obtain shape-preserving properties of the new family of Szász–Mirakyan operators. The symmetry of the properties of the classical Szász–Mirakyan operator and of the properties of the new sequence is investigated. Moreover, we compare this operator with its classical correspondence to show that the new one has superior properties. Finally, some numerical illustrative examples are presented to strengthen our theoretical results. [ABSTRACT FROM AUTHOR]

The class of harmonic convex functions has acquired a very useful and significant placement among the non-convex functions, since this class not only reinforces some major results of the class of convex functions, but also has supported the development of some remarkable results in analysis where the class of convex functions is silent. Therefore, many researchers have deployed themselves to explore valuable results for this class of non-convex functions. This paper obtains new discrete inequalities for univariate harmonic convex functions on linear spaces related to a Jensen-type and a variant of the Jensen-type results. Our results are refinements of very important recent inequalities presented by Dragomir and Baloch et al. Furthermore, we provide the natural applications of our results. [ABSTRACT FROM AUTHOR]

In this paper, firstly, we present an integral identity for functions of two variables via Riemann–Liouville fractional integrals. Then, a Newton-type inequality via partially differentiable coordinated convex mappings is derived by taking the absolute value of the obtained identity. Moreover, several inequalities are obtained with the aid of the Hölder and power mean inequality. In addition, we investigate some Newton-type inequalities utilizing mappings of two variables with bounded variation. Finally, we gave some mathematical examples and their graphical behavior to validate the obtained inequalities. [ABSTRACT FROM AUTHOR]

The calculus in the absence of limits is known as quantum calculus. With a difference operator, it substitutes the classical derivative, which permits dealing with sets of functions that are non-differentiations. The theory of integral inequality in quantum calculus is a field of mathematics that has been gaining considerable attention recently. Despite the fact of its application in discrete calculus, it can be applied in fractional calculus as well. In this paper, some new Anderson type q-integral and h-integral inequalities are given using a Feng Qi integral inequality in quantum calculus. These findings are highly beneficial for basic frontier theories, and the techniques offered by technology are extremely useful for those who can stimulate research interest in exploring mathematical applications. Due to the interesting properties in the field of mathematics, integral inequalities have a tied correlation with symmetric convex and convex functions. There exist strong correlations and expansive properties between the different fields of convexity and symmetric function, including probability theory, convex functions, and the geometry of convex functions on convex sets. The main advantage of these essential inequalities is that they can be converted into time-scale calculus. This kind of inevitable inequality can be very helpful in various fields where coordination plays an important role. [ABSTRACT FROM AUTHOR]

In this paper, we derive some new quantum estimates of generalized Hermite–Hadamard–Jensen–Mercer type of inequalities, essentially using q-differentiable convex functions. With the help of numerical examples, we check the validity of the results. We also discuss some special cases which show that our results are quite unifying. To show the efficiency of our main results, we offer some interesting applications to special means. [ABSTRACT FROM AUTHOR]

Twin extreme learning machine (TELM) is a phenomenon of symmetry that improves the performance of the traditional extreme learning machine classification algorithm (ELM). Although TELM has been widely researched and applied in the field of machine learning, the need to solve two quadratic programming problems (QPPs) for TELM has greatly limited its development. In this paper, we propose a novel TELM framework called Lagrangian regularized twin extreme learning machine (LRTELM). One significant advantage of our LRTELM over TELM is that the structural risk minimization principle is implemented by introducing the regularization term. Meanwhile, we consider the square of the l 2 -norm of the vector of slack variables instead of the usual l 1 -norm in order to make the objective functions strongly convex. Furthermore, a simple and fast iterative algorithm is designed for solving LRTELM, which only needs to iteratively solve a pair of linear equations in order to avoid solving two QPPs. Last, we extend LRTELM to semi-supervised learning by introducing manifold regularization to improve the performance of LRTELM when insufficient labeled samples are available, as well as to obtain a Lagrangian semi-supervised regularized twin extreme learning machine (Lap-LRTELM). Experimental results on most datasets show that the proposed LRTELM and Lap-LRTELM are competitive in terms of accuracy and efficiency compared to the state-of-the-art algorithms. [ABSTRACT FROM AUTHOR]

This paper studies some inclusion properties of some new subclasses of analytic functions in the open symmetric unit disc U that are associated with the Pascal operator. Furthermore, the integral-preserving properties in a sector for these subclasses are also investigated. [ABSTRACT FROM AUTHOR]

In this paper, the third Hankel determinant for the class N of functions convex in one direction is estimated. An analogous problem is solved for a subclass of N consisting of functions with real coefficients. Additionally, this determinant for odd functions in N is estimated. Moreover, similar results are obtained in the relative class M consisting of functions z f ′ (z) , where f ∈ N . The majority of bounds is sharp. [ABSTRACT FROM AUTHOR]

The connection between generalized convexity and symmetry has been studied by many authors in recent years. Due to this strong connection, generalized convexity and symmetry have arisen as a new topic in the subject of inequalities. In this paper, we introduce the concept of interval-valued preinvex functions on the coordinates in a rectangle from the plane and prove Hermite–Hadamard type inclusions for interval-valued preinvex functions on coordinates. Further, we establish Hermite–Hadamard type inclusions for the product of two interval-valued coordinated preinvex functions. These results are motivated by the symmetric results obtained in the recent article by Kara et al. in 2021 on weighted Hermite–Hadamard type inclusions for products of coordinated convex interval-valued functions. Our established results generalize and extend some recent results obtained in the existing literature. Moreover, we provide suitable examples in the support of our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we give the generalized version of the quantum Simpson's and quantum Newton's formula type inequalities via quantum differentiable α , m -convex functions. The main advantage of these new inequalities is that they can be converted into quantum Simpson and quantum Newton for convex functions, Simpson's type inequalities α , m -convex function, and Simpson's type inequalities without proving each separately. These inequalities can be helpful in finding the error bounds of Simpson's and Newton's formulas in numerical integration. Analytic inequalities of this type as well as particularly related strategies have applications for various fields where symmetry plays an important role. [ABSTRACT FROM AUTHOR]

In this paper, we give some correct quantum type Simpson's inequalities via the application of q-Hölder's inequality. The inequalities of this study are compatible with famous Simpson's 1 / 8 and 3 / 8 quadrature rules for four and six panels, respectively. Several special cases from our results are discussed in detail. A counter example is presented to explain the limitation of Hölder's inequality in the quantum framework. [ABSTRACT FROM AUTHOR]

Convex functions are studied very frequently by means of the Hadamard inequality. A symmetric function leads to the generalization of the Hadamard inequality; the Fejér–Hadamard inequality is one of the generalizations of the Hadamard inequality that holds for convex functions defined on a finite interval along with functions which have symmetry about the midpoint of that finite interval. Lately, integral inequalities for convex functions have been extensively generalized by fractional integral operators. In this paper, inequalities of Hadamard type are generalized by using exponentially (α , h-m)-p-convex functions and an operator containing an extended generalized Mittag-Leffler function. The obtained results are also connected with several well-known Hadamard-type inequalities. [ABSTRACT FROM AUTHOR]

Refining and reversing weighted arithmetic-geometric mean inequalities have been studied in many papers. In this paper, we provide some bounds for the differences between the weighted arithmetic and geometric means, using known inequalities. We improve the results given by Furuichi-Ghaemi-Gharakhanlu and Sababheh-Choi. We also give some bounds on entropies, applying the results in a different approach. We explore certain convex or concave functions, which are symmetric functions on the axis t = 1 / 2 . [ABSTRACT FROM AUTHOR]

This study uses Raina's function to obtain a new coordinated p q -integral identity. Using this identity, we construct several new p q -Simpson's type inequalities for generalized convex functions on coordinates. Setting p 1 = p 2 = 1 in these inequalities yields well-known quantum Simpson's type inequalities for coordinated generalized convex functions. Our results have important implications for the creation of post quantum mathematical frameworks. [ABSTRACT FROM AUTHOR]

Recently in the paper [Mediterr. J. Math. 2016, 13, 1535–1553], the authors introduced and studied a new operator which was defined as a convolution of the three popular linear operators, namely the Sǎlǎgean operator, the Ruscheweyh operator and a fractional derivative operator. In the present paper, we consider an operator which is a convolution operator of only two linear operators (with lesser restricted parameters) that yield various well-known operators, defined by a symmetric way, including the one studied in the above-mentioned paper. Several results on the subordination of analytic functions to this operator (defined below) are investigated. Some of the results presented are shown to involve the familiar Appell function and Hurwitz–Lerch Zeta function. Special cases and interesting consequences being in symmetry of our main results are also mentioned. [ABSTRACT FROM AUTHOR]

Numerical approximations of definite integrals and related error estimations can be made using Simpson's rules (inequalities). There are two well-known rules: Simpson's 1 3 rule or Simpson's quadrature formula and Simpson's 3 8 rule or Simpson's second formula. The aim of the present paper is to extend several inequalities that hold for Simpson's 1 3 rule to Simpson's 3 8 rule. More precisely, we prove a weighted version of Simpson's second type inequality and some Simpson's second type inequalities for Lipschitzian, bounded variations, convex functions and the functions that belong to L q . Some applications of the second type Simpson's inequalities relate to approximations of special means and Simpson's 3 8 formula, and moments of random variables are made. [ABSTRACT FROM AUTHOR]

The field of geometric function theory has thoroughly investigated starlike functions concerning symmetric points. The main objective of this work is to derive certain geometric properties, such as the starlikeness of order δ , convexity of order δ , k-starlikeness, k-uniform convexity, lemniscate starlikeness and convexity, exponential starlikeness and convexity, and pre-starlikeness for the Galué type Struve function (GTSF). Furthermore, the conditions for GTSF belonging to the Hardy space are also derived. The results obtained in this work generalize several results available in the literature. [ABSTRACT FROM AUTHOR]

Building upon previous research in conformable fractional calculus, this study introduces a novel identity. Using this identity as a foundation, we derive a set of conformable fractional Milne-type inequalities specifically designed for differentiable convex functions. The obtained results recover some existing inequalities in the literature by fixing some parameters. These novel contributions aim to enrich the analytical tools available for studying convex functions within the realm of conformable fractional calculus. The derived inequalities reflect an inherent symmetry characteristic of the Milne formula, further illustrating the balanced and harmonious mathematical structure within these frameworks. We provide a thorough example with graphical representations to support our findings, offering both numerical insights and visual confirmation of the established inequalities. [ABSTRACT FROM AUTHOR]

Over the past ten years, analytical functions' reputation in the literature and their application have grown. We study some practical issues pertaining to multivalent functions with bounded boundary rotation that associate with the combination of confluent hypergeometric functions and binomial series in this research. A novel subset of multivalent functions is established through the use of convolution products and specific inclusion properties are examined through the application of second order differential inequalities in the complex plane. Furthermore, for multivalent functions, we examined inclusion findings using Bernardi integral operators. Moreover, we will demonstrate how the class proposed in this study, in conjunction with the acquired results, generalizes other well-known (or recently discovered) works that are called out as exceptions in the literature. [ABSTRACT FROM AUTHOR]

The idea of inequality has been extended from the real plane to the complex plane through the notion of subordination introduced by Professors Miller and Mocanu in two papers published in 1978 and 1981. With this notion came a whole new theory called the theory of differential subordination or admissible functions theory. Later, in 2003, a particular form of inequality in the complex plane was also defined by them as dual notion for subordination, the notion of differential superordination and with it, the theory of differential superordination appeared. In this paper, the theory of differential superordination is applied to confluent hypergeometric function. Hypergeometric functions are intensely studied nowadays, the interest on the applications of those functions in complex analysis being renewed by their use in the proof of Bieberbach's conjecture given by de Branges in 1985. Using the theory of differential superodination, best subordinants of certain differential superordinations involving confluent (Kummer) hypergeometric function are stated in the theorems and relation with previously obtained results are highlighted in corollaries using particular functions and in a sandwich-type theorem. An example is also enclosed in order to show how the theoretical findings can be applied. [ABSTRACT FROM AUTHOR]

Since in many particular cases checking directly the conditions from the definitions of starlikeness or convexity of a function can be difficult, in this paper we use the theory of differential subordination and in particular the method of admissible functions in order to determine conditions of starlikeness and convexity for the confluent (Kummer) hypergeometric function of the first kind. Having in mind the results obtained by Miller and Mocanu in 1990 who used a , c ∈ R , for the confluent (Kummer) hypergeometric function, in this investigation a and c complex numbers are used and two criteria for univalence of the investigated function are stated. An example is also included in order to show the relevance of the original results of the paper. [ABSTRACT FROM AUTHOR]

Let C 0 (h) be a subclass of analytic and close-to-convex functions defined in the open unit disk by the formula R e { (1 − z 2) f ′ (z) } > 0 . In this paper, some coefficient problems for C 0 (h) are considered. Some properties and bounds of several coefficient functionals for functions belonging to this class are provided. The main aim of this paper is to find estimates of the difference and of sum of successive coefficients, bounds of the sum of the first n coefficients and bounds of the n-th coefficient. The obtained results are used to determine coefficient estimates for both functions convex in the imaginary-axis direction with real coefficients and typically real functions. Moreover, the sum of the first initial coefficients for functions with a positive real part and with a fixed second coefficient is estimated. [ABSTRACT FROM AUTHOR]

Recently, two classes of univalent functions S e * and K e were introduced and studied. A function f is in S e * if it is analytic in the unit disk, f (0) = f ′ (0) − 1 = 0 and z f ′ (z) f (z) ≺ e z . On the other hand, g ∈ K e if and only if z g ′ ∈ S e * . Both classes are symmetric, or invariant, under rotations. In this paper, we solve a few problems connected with the coefficients of functions in these classes. We find, among other things, the estimates of Hankel determinants: H 2 , 1 , H 2 , 2 , H 3 , 1 . All these estimates improve the known results. Moreover, almost all new bounds are sharp. The main idea used in the paper is based on expressing the discussed functionals depending on the fixed second coefficient of a function in a given class. [ABSTRACT FROM AUTHOR]

We consider a newly introduced integral operator that depends on an analytic normalized function and generalizes many other previously studied operators. We find the necessary conditions that this operator has to meet in order to preserve convex meromorphic functions. We know that convexity has great impact in the industry, linear and non-linear programming problems, and optimization. Some lemmas and remarks helping us to obtain complex functions with positive real parts are also given. [ABSTRACT FROM AUTHOR]