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In this paper, we study classes of meromorphic harmonic functions defined by Ruscheweyh derivatives. In addition to finding certain analytic criteria, we obtain radii of starlikeness and convexity, and some topological properties for the defined classes of functions. Some applications of these results are also given. [ABSTRACT FROM AUTHOR]

Recently, El-Deeb and Cotîrlă (Mathematics 11:11234834, 2023) used the error function together with a q-convolution to introduce a new operator. By means of this operator the following class R α , ϒ λ , q (δ , η) of analytic functions was studied: R α , ϒ λ , q (δ , η) : = { F : ℜ ((1 − δ + 2 η) H ϒ λ , q F (ζ) ζ + (δ − 2 η) (H ϒ λ , q F (ζ)) ′ + η ζ (H ϒ λ , q F (ζ)) ″) } > α (0 ≦ α < 1). For these general analytic functions F ∈ R β , ϒ λ , q (δ , η) , we give upper bounds for the Fekete–Szegö functional and for the second and third Hankel determinants. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce two new subclasses R sin m (ℑ) and R sin (ℑ) of analytic functions by making use of subordination involving the sine function and the modified sigmoid activation function ℑ (v) = 2 1 + e − v , v ≥ 0 in the open unit disc E. Our purpose is to obtain some initial coefficients, Fekete–Szego problems, and upper bounds for the third- and fourth-order Hankel determinants for the functions belonging to these two classes. All the bounds that we will find here are sharp. We also highlight some known consequences of our main results. [ABSTRACT FROM AUTHOR]

This article comprises the study of class S E ∗ that represents the class of normalized analytic functions f satisfying ς f ′ (z) / f (ς) ≺ sec h (ς) . The geometry of functions of class S E ∗ is star-shaped, which is confined in the symmetric domain of a secant hyperbolic function. We find sharp coefficient results and sharp Hankel determinants of order two and three for functions in the class S E ∗ . We also investigate the same sharp results for inverse coefficients. [ABSTRACT FROM AUTHOR]

In continuation of Zayed and Bulboacă work in (J. Inequal. Appl. 2022:158, 2022), this paper discusses the geometric characterization of the normalized form of the generalized Bessel function defined by V ρ , r (z) : = z + ∑ k = 1 ∞ (− r) k 4 k (1) k (ρ) k z k + 1 , z ∈ U , for ρ , r ∈ C ∗ : = C ∖ { 0 } . Precisely, we will use a sharp estimate for the Pochhammer symbol, that is, Γ (a + n) / Γ (a + 1) > (a + α) n − 1 , or equivalently (a) n > a (a + α) n − 1 , that was firstly proved by Baricz and Ponnusamy for n ∈ N ∖ { 1 , 2 } , a > 0 and α ∈ [ 0 , 1.302775637 ... ] in (Integral Transforms Spec. Funct. 21(9):641–653, 2010), and then proved in our paper by another method to improve it using the partial derivatives and the two-variable functions' extremum technique for n ∈ N ∖ { 1 , 2 } , a > 0 and 0 ≤ α ≤ 2 , and used to investigate the orders of starlikeness and convexity. We provide the reader with some examples to illustrate the efficiency of our theory. Our results improve, complement, and generalize some well-known (nonsharp) estimates, as seen in the Concluding Remarks and Outlook section. [ABSTRACT FROM AUTHOR]

The Ruscheweyh derivative operator is used in this paper to introduce and investigate interesting general subclasses of the function class Σ m of m-fold symmetric bi-univalent analytic functions. Estimates of the initial Taylor-Maclaurin coefficients | a m + 1 | and | a 2 m + 1 | are obtained for functions of the subclasses introduced in this study, and the consequences of the results are discussed. Additionally, the Fekete-Szegö inequalities for these classes are investigated. The results presented could generalize and improve some recent and earlier works. In some cases, our estimates are better than the existing coefficient bounds. Furthermore, within the engineering domain, the utilization of the Ruscheweyh derivative operator can encompass a broad spectrum of engineering applications, including the robotic manipulation control, optimizing optical systems, antenna array signal processing, image compression, and control system filter design. It emphasizes the potential for innovative solutions that can significantly enhance the reliability and effectiveness of engineering applications. [ABSTRACT FROM AUTHOR]

The present investigation aims to examine the geometric properties of the normalized form of the combination of generalized Lommel–Wright function J λ , μ ν , m (z) : = Γ m (λ + 1) Γ (λ + μ + 1) 2 2 λ + μ z 1 − (ν / 2) − λ J λ , μ ν , m (z) , where the function J λ , μ ν , m satisfies the differential equation J λ , μ ν , m (z) : = (1 − 2 λ − ν) J λ , μ ν , m (z) + z (J λ , μ ν , m (z)) ′ with J ν , λ μ , m (z) = (z 2) 2 λ + ν ∑ k = 0 ∞ (− 1) k Γ m (k + λ + 1) Γ (k μ + ν + λ + 1) (z 2) 2 k for λ ∈ C ∖ Z − , Z − : = { − 1 , − 2 , − 3 , ... } , m ∈ N , ν ∈ C , and μ ∈ N 0 : = N ∪ { 0 } . In particular, we employ a new procedure using mathematical induction, as well as an estimate for the upper and lower bounds for the gamma function inspired by Li and Chen (J. Inequal. Pure Appl. Math. 8(1):28, 2007), to evaluate the starlikeness and convexity of order α, 0 ≤ α < 1 . Ultimately, we discuss the starlikeness and convexity of order zero for J λ , μ ν , m , and it turns out that they are useful to extend the range of validity for the parameter λ to λ ≥ 0 where the main concept of the proofs comes from some technical manipulations given by Mocanu (Libertas Math. 13:27–40, 1993). Our results improve, complement, and generalize some well-known (nonsharp) estimates. [ABSTRACT FROM AUTHOR]

In recent years, many subclasses of univalent functions, directly or not directly related to the exponential functions, have been introduced and studied. In this paper, we consider the class of S e ∗ for which z f ′ (z) / f (z) is subordinate to e z in the open unit disk. The classic concept of Hankel determinant is generalized by replacing the inverse logarithmic coefficient of functions belonging to certain subclasses of univalent functions. In particular, we obtain the best possible bounds for the second Hankel determinant of logarithmic coefficients of inverse starlike functions subordinated to exponential functions. This work may inspire to pay more attention to the coefficient properties with respect to the inverse functions of various classes of univalent functions. [ABSTRACT FROM AUTHOR]

Two new subclasses of the class of bi-Bazilevič functions, which are related to the Fibonacci-number series and the square-root functions, are introduced and studied in this article. Under a special choice of the parameter involved, these two classes of Bazilevič functions reduce to two new subclasses of star-like biunivalent functions related with the Fibonacci-number series and the square-root functions. Using the Faber polynomial expansion (FPE) technique, we find the general coefficient bounds for the functions belonging to each of these classes. We also find bounds for the initial coefficients for bi-Bazilevič functions and demonstrate how unexpectedly these initial coefficients behave in relation to the square-root functions and the Fibonacci-number series. [ABSTRACT FROM AUTHOR]

Due to the major importance of the study of the logarithmic coefficients for univalent functions, in this paper we find the sharp upper bounds for some expressions associated with logarithmic coefficients of functions that belong to some well-known classes of analytic functions in the open unit disk of the complex plane. [ABSTRACT FROM AUTHOR]

In the present paper, we apply the theory of differential subordination to extend the famous Jack lemma for analytic functions. Also, using this extension we will produce new versions of the open-door lemma for analytic functions, and utilizing them some simple conditions for starlikeness of analytic and meromorphic functions are obtained. Our results improve and extend the earlier results investigated in the literature. [ABSTRACT FROM AUTHOR]

The objective of this paper is to introduce new classes of m-fold symmetric bi-univalent functions. We discuss estimates on the Taylor–Maclaurin coefficients | a m + 1 | and | a 2 m + 1 | , and the Fekete–Szegő problem is also considered for the new classes of functions introduced. We denote these classes by M F − S Σ , m p , q (h) , M F − S Σ , m p , q (s) , and M F − S Σ , m b , d . Quantum calculus aspects are also considered in this study to enhance its novelty and to obtain more interesting results. [ABSTRACT FROM AUTHOR]

Our purpose in the present investigation is to study certain geometric properties such as the close-to-convexity, convexity, and starlikeness of the hypergeometric function z 1 F 2 (a ; b , c ; z) in the open unit disk. The usefulness of the main results for some familiar special functions like the modified Sturve function, the modified Lommel function, the modified Bessel function, and the F 1 0 (− ; c ; z) function are also mentioned. We further consider a boundedness property of the function F 2 1 (a ; b , c ; z) in the Hardy space of analytic functions. Several corollaries and special cases of the main results are also pointed out. [ABSTRACT FROM AUTHOR]

The normalization of the generalized Bessel functions U σ , r (σ , r ∈ C) defined by U σ , r (z) = z + ∑ j = 1 ∞ (− r) j 4 j (1) j (σ) j z j + 1 was introduced, and some of its geometric properties have been presented previously. The main purpose of the present paper is to complete the results given in the literature by employing a new procedure. We first used an identity for the logarithmic of the gamma function as well as an inequality for the digamma function to establish sufficient conditions on the parameters so that U σ , r is starlike or convex of order α (0 ≤ α ≤ 1) in the open unit disk. Moreover, the starlikeness and convexity of U σ , r have been considered where the leading concept of the proofs comes from the starlikeness of the power series f (z) = ∑ j = 1 ∞ A j z j and the classical Alexander theorem between the classes of starlike and convex functions. We gave a simple proof to show that our conditions are not contradictory. Ultimately, the close-to-convexity of (z cos z) ∗ U σ , r and (sin z) ∗ U σ , r (z 2) z have been determined, where "∗" stands for the convolution between the power series. [ABSTRACT FROM AUTHOR]

The normalization of the combination of generalized Lommel–Wright function J κ 1 , κ 2 κ 3 , m (z) (m ∈ N , κ 3 > 0 and κ 1 , κ 2 ∈ C ) defined by J κ 1 , κ 2 κ 3 , m (z) : = Γ m (κ 1 + 1) Γ (κ 1 + κ 2 + 1) 2 2 κ 1 + κ 2 z 1 − (κ 2 / 2) − κ 1 J κ 1 , κ 2 κ 3 , m (z) , where J κ 1 , κ 2 κ 3 , m (z) : = (1 − 2 κ 1 − κ 2) J κ 1 , κ 2 κ 3 , m (z) + z (J κ 1 , κ 2 κ 3 , m (z)) ′ and J κ 1 , κ 2 κ 3 , m (z) = (z 2) 2 κ 1 + κ 2 ∑ n = 0 ∞ (− 1) n Γ m (n + κ 1 + 1) Γ (n κ 3 + κ 1 + κ 2 + 1) (z 2) 2 n , was previously introduced and some of its geometric properties have been considered. In this paper, we report conditions for J κ 1 , κ 2 κ 3 , m (z) to be starlike and convex of order α, 0 ≤ α < 1 , inside the open unit disk using some technical manipulations of the gamma and digamma functions, as well as inequality for the digamma function that has been proved (Guo and Qi in Proc. Am. Math. Soc. 141(3):1007–1015, 2013). In addition, a method presented by Lorch (J. Approx. Theory 40(2):115–120 1984) and further developed by Laforgia (Math. Compet. 42(166):597–600 1984) is applied to establish firstly sharp inequalities for the shifted factorial that will be used to obtain the order of starlikeness and convexity. We compare then the obtained orders of starlikeness and convexity with some important consequences in the literature as well as the results proposed by all techniques to demonstrate the accuracy of our approach. Ultimately, a lemma by (Fejér in Acta Litt. Sci. 8:89–115 1936) is used to prove that the modified form of the function J κ 1 , κ 2 κ 3 , m (z) defined by I κ 1 , κ 2 κ 3 , m (z) = J κ 1 , κ 2 κ 3 , m (z) ∗ z / (1 + z) is in the class of starlike and convex functions, respectively. Further work regarding the function J κ 1 , κ 2 κ 3 , m (z) is underway and will be presented in a forthcoming paper. [ABSTRACT FROM AUTHOR]

The scope of our investigation is to study the geometric properties of the normalized form of the combination of generalized Lommel–Wright function J ν , λ μ , m defined by J ν , λ μ , m (z) : = Γ m (λ + 1) Γ (λ + ν + 1) 2 2 λ + ν z 1 − (ν / 2) − λ I ν , λ μ , m (z) , where I ν , λ μ , m (z) : = (1 − 2 λ − ν) J ν , λ μ , m (z) + z (J ν , λ μ , m (z)) ′ and J ν , λ μ , m (z) = (z 2) 2 λ + ν ∑ n = 0 ∞ (− 1) n Γ m (n + λ + 1) Γ (n μ + ν + λ + 1) (z 2) 2 n , with m ∈ N , μ > 0 and λ , ν ∈ C , including starlikeness and convexity of order α (0 ≤ α < 1 ) in the open unit disc using the two-sided inequality for the Fox–Wright functions that has been proved by Pogány and Srivastava in (Comput. Math. Appl. 57(1):127–140, 2009). Further, the orders of starlikeness and convexity are also evaluated using some classical tools. We then compare the orders of starlikeness and convexity given by both techniques to illustrate the efficacy of the approach. In addition, we proved that for some values of α, if λ > − 1 then Re (J ν , λ μ , m (z) / z) > α , z ∈ U , and if λ ≥ (10 − 6) / 4 then the function (J ν , λ μ , m (z 2) / z) ∗ sin z is close-to-convex with respect to 1 / 2 log ((1 + z) / (1 − z)) where ∗ stands for the Hadamard product (or convolution) of two power series. [ABSTRACT FROM AUTHOR]

Bieberbach's conjecture was very important in the development of geometric function theory, not only because of the result itself, but also due to the large amount of methods that have been developed in search of its proof. It is in this context that the integral transformations of the type f α (z) = ∫ 0 z (f (ζ) / ζ) α d ζ or F α (z) = ∫ 0 z (f ′ (ζ)) α d ζ appear. In this note we extend the classical problem of finding the values of α ∈ C for which either f α or F α are univalent, whenever f belongs to some subclasses of univalent mappings in D , to the case of logharmonic mappings by considering the extension of the shear construction introduced by Clunie and Sheil-Small in (Clunie and Sheil-Small in Ann. Acad. Sci. Fenn., Ser. A I 9:3–25, 1984) to this new scenario. [ABSTRACT FROM AUTHOR]

A number of families of q-extensions of analytic functions in the open unit disk U have been defined by means of basic (or q-)calculus and considered from many distinctive prospectives and viewpoints. In this paper, we generalize and study certain subclasses of analytic functions involving higher-order q-derivative operators. We settle characteristic equations for these presumably new classes and also study numerous coefficient inequalities. For the results obtained in this presentation, we also carry out appropriate connections with those in multiple other concerning works on this subject. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to study radii problems for two general classes including various known subclasses of analytic functions associated with the normalized form of Legendre polynomials of odd degree. We also obtain some special cases of the main results presented here with some useful examples. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and investigate new classes of normalized analytic functions in an open unit disk with bounded radius and bounded boundary rotation by using the subordination. We discuss inclusion results, co-efficient bounds, growth and distortion theorems of the classes. Moreover, we compute the radii of strong starlikeness, convexity and starlikeness of the classes. It is interesting to mention that most of our findings are best possible as compared to the existing results in the literature. [ABSTRACT FROM AUTHOR]

Let ψ μ , ν (z) = (1 − 2 cos ν e i μ z + e 2 i μ z 2) − 1 , μ , ν ∈ [ 0 , 2 π) and p be an analytic mapping with Re p > 0 on the open unit disk. We consider the sense-preserving planar harmonic mappings f = h + g ‾ , which are shears of the mapping ∫ 0 z ψ μ , ν (ξ) p (ξ) d ξ in the direction μ. These mappings include the harmonic right half-plan mappings, vertical strip mappings, and their rotations. For various choices of dilatations g ′ / h ′ of f, sufficient conditions are found for the convex combinations of these mappings to be univalent and convex in the direction μ. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some potentially useful conditions (or criteria) for the Carathéodory functions as a certain class of analytic functions by applying Nunokawa's lemma. We also obtain several conditions for strong starlikeness and close-to-convexity as special cases of the main results presented here. [ABSTRACT FROM AUTHOR]

The analytic functions, mapping the open unit disk onto petal and oval type regions, introduced by Noor and Malik (Comput. Math. Appl. 62:2209-2217, 2011), are considered to define and study their associated close-to-convex functions. This work includes certain geometric properties like sufficiency criteria, coefficient estimates, arc length, the growth rate of coefficients of Taylor series, integral preserving properties of these functions. [ABSTRACT FROM AUTHOR]

Complex-valued harmonic functions that are univalent and sense preserving in the open unit disk can be written in the form $f=h+\overline{g}$, where h and g are analytic. In this paper we investigate some classes of univalent harmonic functions with varying coefficients related to Janowski functions. By using the extreme points theory we obtain necessary and sufficient convolution conditions, coefficients estimates, distortion theorems, and integral mean inequalities for these classes of functions. The radii of starlikeness and convexity for these classes are also determined. [ABSTRACT FROM AUTHOR]

We investigate several sufficient conditions on a function to be convex in one direction or starlike in one direction. [ABSTRACT FROM AUTHOR]

In this paper we use a result of Nunokawa to extend some results on univalent functions given by Miller and Mocanu. As a consequence, we get several sufficient conditions for starlikeness over the expression $f(z)f''(z)/f^{\prime2}(z)$. [ABSTRACT FROM AUTHOR]

We prove some new sufficient conditions for a function to be p-valent or p-valently starlike in the unit disc. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of analytic functions defined by a new convolution operator $J_{(\lambda_{p}),(\mu _{q}),b}^{s,a,\lambda}$ which generalizes the well-known Srivastava-Attiya operator investigated by Srivastava and Attiya (Integral Transforms Spec. Funct. 18:207-216, 2007). We derive coefficient inequalities, distortion theorems, extreme points and the Fekete-Szegö problem for this new function class. [ABSTRACT FROM AUTHOR]

In the present paper we investigate a class of harmonic mappings for which the second dilatation is a close-to-convex function of complex order b, $b\in\mathbb{C} / \{ 0 \}$ (Lashin in Indian J. Pure Appl. Math. 34(7):1101-1108, 2003). [ABSTRACT FROM AUTHOR]

Let $\mathcal{A}_{n}$ ( $n\in\mathbb{N}$) be the class of certain analytic functions $f(z)$ in the open unit disk $\mathbb{U}$ and $\mathcal{P}_{n}(\lambda)$ be the subclass of $\mathcal{A}_{n}$ consisting of $f(z)$ which satisfy $|f''(z)| \leqq \lambda$ ( $\lambda> 0$) in $\mathbb{U}$. Some properties for the class $\mathcal{P}_{n}(\lambda)$, which are the improvements of the previous results due to Ponnusamy and Singh (Complex Var. Theory Appl. 34:276-291, 1997), are discussed. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to introduce a generalized hypergeometric distribution and obtain some necessary and sufficient conditions for generalized hypergeometric distribution series belonging to certain classes of univalent functions associated with the conic domains. We also investigate some inclusion relations. Finally, we discuss an integral operator related to this series. [ABSTRACT FROM AUTHOR]

The conjecture proposed by Raina and Sokòł [Hacet. J. Math. Stat. 44(6):1427–1433 (2015)] for a sharp upper bound on the fourth coefficient has been settled in this manuscript. An example is constructed to show that their conjectures for the bound on the fifth coefficient and the bound related to the second Hankel determinant are false. However, the correct bound for the latter is stated and proved. Further, a sharp bound on the initial coefficients for normalized analytic function f such that z f ′ (z) / f (z) ≺ 1 + λ z , λ ∈ (0 , 1 ] , have also been obtained, which contain many existing results. [ABSTRACT FROM AUTHOR]

We consider a family of analytic and normalized functions with the property that z f ′ (z) / f (z) (or 1 + z f ″ (z) / f ′ (z) ) lies in a domain bounded by a right branch of a hyperbola ρ = ρ (s) = (2 cos φ s) − s , where 0 < s ≤ 1 and | φ | < (π s) / 2 . A comprehensive characteristic of that families and relations with the well known families of univalent functions are presented. Some relevant examples are indicated. [ABSTRACT FROM AUTHOR]

Using the third-order differential subordination basic results, we introduce certain classes of admissible functions and investigate some applications of third-order differential subordination for p-valent functions associated with generalized fractional differintegral operator. [ABSTRACT FROM AUTHOR]

In the current work, we delineate a comprehensive class of bi-univalent functions connected with a bounded boundary variation to get the estimates of the first two Taylor–Maclaurin coefficients. In addition, certain special cases and some appealing interpretation of the results presented here are pointed out. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new subclass of p-harmonic functions and investigate the univalence and sense-preserving, extreme points, distortion bounds, convex combination, neighborhoods of mappings belonging to the subclass. Relevant connections of the results presented here with the results of previous research are briefly indicated. Finally, we also prove new properties of the Hadamard product of these classes. [ABSTRACT FROM AUTHOR]

In this paper, some geometric properties of normalized Mittag-Leffler functions are investigated. We focus on starlikeness of order 2 μ + η − 1 and convexity in the direction of imaginary axis. In addition, we study pre-starlikeness of Mittag-Leffler functions. The results are obtained by using the positivity technique. [ABSTRACT FROM AUTHOR]

In this paper, the authors introduce a new subclass of meromorphic q-starlike functions which are associated with the Janowski functions. A characterization of meromorphically q-starlike functions associated with the Janowski functions has been obtained when the coefficients in their Laurent series expansion about the origin are all positive. This leads to a study of coefficient estimates, distortion theorems, partial sums, and the radius of starlikeness estimates for this class. It is seen that the class considered demonstrates, in some respects, properties analogous to those possessed by the corresponding class of univalent analytic functions with negative coefficients. [ABSTRACT FROM AUTHOR]

In this work, we introduce certain subclasses of analytic functions involving the integral operators that generalize the class of uniformly starlike, convex, and close-to-convex functions with respect to symmetric points. We then establish various inclusion relations for these newly defined classes. [ABSTRACT FROM AUTHOR]

The object of the present paper is to investigate various conditions for Carathéodory functions in the open unit disk. Also we give some applications to univalent functions as special cases. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class k-US(q,γ,m,p), γ∈C∖{0}, of multivalent functions using a newly defined q-analogue of a Salagean type differential operator. We investigate the coefficient problem, Fekete-Szego inequality, and some other properties related to subordination. Relevant connections of the results presented here with those obtained in the earlier work are also pointed out. [ABSTRACT FROM AUTHOR]

In the present paper, we investigate majorization properties for the class Mβα(p,γ) of uniformly starlike functions and the class Nβα(p,θ) of spiral-like functions related to an exponential function, which are defined through the Liu-Owa integral operator Qβ,pα given by (1.5). Also, some special cases of our main results in a form of corollaries are shown. [ABSTRACT FROM AUTHOR]