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The paper refers to an aggregated estimate of two initial terms in the power series expansions of holomorphic functions and mappings of several complex variables. The authors solve first this problem in some Bavrin’s families of functions, i.e., functions f : G → C , holomorphic in bounded complete n-circular domains G ⊂ C n and satisfying conditions, similar as in geometric function theory of one variable. Next, they apply this result to a family of mappings F : B n → C n biholomorphic in the open unit Euclidean ball B n ⊂ C n. [ABSTRACT FROM AUTHOR]

The Fekete-Szegö inequality is one of the inequalities for the coefficients which associated with the famous Bieberbach conjecture. Other issues associated with this inequality are determining the Hankel determinant denoted as Hd inequalities which are involved in the study of the singularities and integral coefficients in the Taylor series representations. In this investigation, we discuss such inequalities for certain mappings g for which \[ (g^{\prime}(z))^{\alpha}\left[ \frac{2zg^{\prime}(z)}{g(z)-g(-z)}\right] ^{(1-\alpha)}\prec1+\sin z,(0\leq \alpha \leq1) \] (g ′ (z)) α [ 2 z g ′ (z) g (z) − g (− z) ] (1 − α) ≺ 1 + sin ⁡ z , (0 ≤ α ≤ 1) lies in the image domain starlike about 1 and symmetric about real axis. Particularly, we obtain certain inequality for functions involving Poisson distribution. Furthermore, we also find the second Hankel determinant and other inequalities related to this newly defined class. [ABSTRACT FROM AUTHOR]

This article presents new classes of ζ-uniformly q-starlike and q-convex analytic functions of order υ using the principle of q-calculus. The renowned classes of starlike and convex functions are utilized to establish these classes. The investigation of these uniformly classes leads to the study of some geometric notions, including coefficient estimates, sharpness, distortion and growth theorems, and convex linear combinations. Furthermore, some subordination properties involving integral means inequalities and subordinate sequences are examined. [ABSTRACT FROM AUTHOR]

Special functions and special polynomials have been used and studied widely in the context of Geometric function theory of Complex Analysis by the many authors. Here, in our present investigations, making use of the convolution, we first introduce a new subclass of Bazilevivč bi-univalent functions involving the Bernoulli Polynomials. We then find the first two Taylor-Maclaurin coefficient | a 2 | and | a 3 | for our defined functions class. We then calculate the bounds for the Fekete Szegö inequality for the defined function class. Also some known consequences of our main results are highlighted in the form of Corollaries. [ABSTRACT FROM AUTHOR]

In this article, we study the majorization problem for the general class of functions with bounded radius rotation, which the authors introduced here. Furthermore, the coefficient bound for majorized functions associated with this class is derived. Consequently, we present new results as corollaries and point out relevant connections between the main results obtained from the ones in the literature. [ABSTRACT FROM AUTHOR]

We introduce a primitive class of analytic functions, by specializing in many well-known classes, classify Ma-Minda functions based on its conditions and their interesting geometrical aspects. Further, study a newly defined subclass of starlike functions involving a special type of Ma-Minda function introduced here for obtaining inclusion and radius results. We also establish some majorization, Bloch function norms, and other related problems for the same class. [ABSTRACT FROM AUTHOR]

Let ℌ denote the class of complex-valued harmonic functions f defined in the open unit disc D and normalized by f(0) = fz(0) − 1 = 0. In this paper, we define a new generalized subclass of ℌ associated with the (p, q)–Ruscheweyh-type harmonic differential operator in D . We first obtain a sufficient coefficient condition that guarantees that a function f in ℌ is sense-preserving harmonic univalent in D and belongs to the aforementioned class. Using this coefficient condition, we then examine ratios of partial sums of f in ℌ. In all cases the results are sharp. In addition, the results so obtained generalize the related works of some authors, and many other new results are obtained. [ABSTRACT FROM AUTHOR]

A normalized analytic function defined on the open unit disk is a bounded turning function if its derivative has positive real part. Such functions are univalent, and therefore, we find sufficient conditions for a function to be a bounded turning function. In this paper, we prove a general differential subordination theorem in terms of the derivative, the pre-Schwarzian derivative, and the Schwarzian derivative, providing sufficient conditions for a function to be a bounded turning function. We then apply the result to obtain several simple sufficient conditions. [ABSTRACT FROM AUTHOR]

In the present paper with the aid of subordination, the authors introduce two subclasses of analytic functions denoted by S α , β (λ) (α , β , λ ∈ R , α < 1 , β > 1 , λ ≥ 0) and G (λ) defined in the open unit disk D : = { z ∈ C : | z | < 1 } . These subclasses are defined through a certain univalent function S α , β and the generating function of the Gregory coefficients G (λ) . We determine upper bounds of the initial coefficients, Fekete–Szeg o ¨ functional, Hankel determinant of second order, logarithmic coefficients and inverse coefficients of the functions belongs to these subclasses. Some of the corollaries of the main results are also pointed out. [ABSTRACT FROM AUTHOR]

In this paper, we first establish a refinement of the coefficient inequality for subordinate functions on the unit disc U in ℂ . Next, as applications of this inequality, we will obtain some refinements of the Fekete and Szegő inequalities for a class of holomorphic mappings associated with starlike mappings and quasi-convex mappings of type B on the unit ball B of a complex Banach space. The results presented here would generalize and improve some recent works of several authors [12, 20, 23]. [ABSTRACT FROM AUTHOR]

The present study aims to investigate a majorization problem for a class of q-starlike functions and for the subclass S ∗ (χ) of starlike functions as well as to rectify certain previous results. For this goal, the valid form connected with Nehari's inequality for q-difference operators was presented. Further, our findings contribute to the novelty of the results of this topic by revealing new insights. In addition, our results with previous works have connected, drawing inspiration from existing papers. Also, several new implications of the main result expressed as corollaries were provided and advances in the studied topic were made. [ABSTRACT FROM AUTHOR]

In this paper we study the asymptotic behavior of the radii of starlikeness of the normalized Jackson’s second and third q-Bessel functions, focusing on their large orders. To achieve this, we use the Rayleigh sums of positive zeros of both q-Bessel functions, and determine the coefficients of the asymptotic expansions. We derive complete asymptotic expansions for these radii of starlikeness and provide recurrence relations for the coefficients of these expansions. The proofs rely on the notion of Rayleigh sums of positive zeros of q-Bessel functions, Kvitsinsky’s results on spectral zeta functions for q-Bessel functions and asymptotic inversion. Moreover, we derive bounds for the radius of starlikeness of q-Bessel functions by using potential polynomials. [ABSTRACT FROM AUTHOR]

In this study, subordination, superordination, and sandwich theorems are established for a class of p-valent analytic functions involving a generalized integral operator that has as a special case p-valent Sălăgean integral operator. Relevant connections of the new results with several well-known ones are given as a conclusion for this investigation.

Special functions and special polynomials have been used and studied widely in the context of Geometric function theory of Complex Analysis by the many authors. Here, in our present investigations, making use of the convolution, we first introduce a new subclass of Bazilevivč bi-univalent functions involving the Bernoulli Polynomials. We then find the first two Taylor-Maclaurin coefficient [Formula: see text] and [Formula: see text] for our defined functions class. We then calculate the bounds for the Fekete Szegö inequality for the defined function class. Also some known consequences of our main results are highlighted in the form of Corollaries.

In this article, we study the majorization problem for the general class of functions with bounded radius rotation, which the authors introduced here. Furthermore, the coefficient bound for majorized functions associated with this class is derived. Consequently, we present new results as corollaries and point out relevant connections between the main results obtained from the ones in the literature.

Let R α , β (z) = z + ∑ n = 2 ∞ β n - 1 Γ (1 + α) Γ ((1 + α) n) z n be the normalized Rabotnov functions. The purpose of the present paper is to determine necessary and sufficient conditions and inclusion relation for the normalized Rabotnov function R α , β (z) to be in two subclasses of analytic functions with positive coefficients. Further, we consider an integral operator related to the Rabotnov function R α , β (z) . Several examples of the main results are also considered. [ABSTRACT FROM AUTHOR]

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]

In this paper, we obtain sharp bounds for the second Hankel determinant of logarithmic coefficients H2,1(Ff/2) of bounded turning functions of order α. Furthermore, we obtain sharp bounds of the second Hankel determinant of logarithmic inverse coefficients of H2,1(Ff−1/2), where f-1 is the inverse function of f. For the class 풫′(α), we conclude that the bounds for H2,1(Ff /2) and H2,1(Ff−1/2) are the same only for 1/4 ≤ α < 1. [ABSTRACT FROM AUTHOR]

In this paper, we determine sharp bounds on some Hankel determinants involving initial coefficients, inverse coefficients, and logarithmic inverse coefficients for two associated subclasses of Sakaguchi functions related to the lemniscate of Bernoulli and exponential function. Further, we compute sharp bounds on the second-order Hermitian–Toeplitz determinants involving logarithmic coefficients and logarithmic inverse coefficients for such subclasses. We also discuss the invariance property for the obtained estimates with respect to various coefficients. [ABSTRACT FROM AUTHOR]

In this paper we define a Bernardi type quantum integral operator. It transforms the starlike univalent in the unit disk into a starlike region in it. We show that the upper-bound of the third-order Hankel determinant for classes of q-starlike functions is connected with a q-analogue integral operator, defined by a modified q-Bernardi integral operator. The Fekete-Szegö inequality of these classes is also investigated. Numerous well-known specific instances, examples and graphics are listed in the paper. The computations are done by Mathematica 13.3. [ABSTRACT FROM AUTHOR]

In this paper, we first obtain a refined version of the Bohr inequality of norm-type for holomorphic mappings with lacunary series on the polydisk in $ \mathbb {C}^n $ C n under some restricted conditions. Next, we determine the refined version of the Bohr inequality for holomorphic functions defined on a balanced domain G of a complex Banach space X and take values in the unit disk $ \mathbb {D} $ D . Furthermore, as a consequence of one of these results, we obtain a refined version of the Bohr-type inequality for harmonic functions $ f=h+\bar {g} $ f = h + g ¯ defined on a balanced domain $ G\subset X $ G ⊂ X. All the results are proved to be sharp. [ABSTRACT FROM AUTHOR]

Odd univalent analytic functions played an instrumental role in the proof of the celebrated Bieberbach conjecture. In this article, we explore odd univalent harmonic mappings, focusing on coefficient estimates, growth and distortion theorems. Motivated by the unresolved harmonic analogue of the Bieberbach conjecture, we investigate specific subclasses of S H 0 , the class of sense-preserving univalent harmonic functions. We provide sharp coefficient bounds for functions exhibiting convexity in one direction and extend our findings to a more generalized class including the major geometric subclasses of S H 0 . Additionally, we analyze the inclusion of these functions in Hardy spaces and broaden the range of p for which they belong. In particular, the results of this article enhance understanding and highlight analogous growth patterns between odd univalent harmonic functions and the harmonic Bieberbach conjecture. We conclude the article with 2 conjectures and possible scope for further study as well. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the presence of positive solutions for system of fractional q-differential inclusions involving sequential derivatives with respect to the p-Laplacian operator. By using fixed point technique we obtain a new solution for inclusion or boundary value problems with special integral and derivative conditions. At the end, we give an example to show the effect of the solution of this device. The conclusion is expressed to introduce future works. [ABSTRACT FROM AUTHOR]

In this article, we investigate meromorphic univalent harmonic functions having a simple pole at the origin. First we establish sufficient conditions for the univalence of function f within the broader class of meromorphic harmonic functions. Then, we derive coefficient estimate for some geometric subclasses of meromorphic univalent harmonic functions. Subsequently, we provide several necessary and sufficient conditions for f to be hereditarily λ -spirallike. Finally, we offer a comprehensive characterization of hereditarily meromorphic harmonic Archimedean and hyperbolic spirallike functions. [ABSTRACT FROM AUTHOR]

We determine the sharp bounds on the second Hankel determinants of logarithmic coefficients and the third Hankel determinants for Ozaki close-to-convex functions. [ABSTRACT FROM AUTHOR]

This article focuses on the improvement of the classic Bohr’s inequality for bounded analytic functions on the unit disk. We give some sharp versions of Bohr’s inequality, generalizing the previous results. [ABSTRACT FROM AUTHOR]

Using the Hadamard factorization, the exponential radii of starlikeness and convexity for various special functions like Wright function, Lommel function, Struve function, Ramanujan type entire function, cross product and product of Bessel function have been investigated. For certain ranges of the parameters appearing in these special functions, the precise values of the exponential radii of starlikeness and convexity are calculated as the solutions of transcendental equations. The interlacing property of the zeros of special functions and their derivatives is the fundamental technique utilized to demonstrate these results. [ABSTRACT FROM AUTHOR]

For normalized starlike functions f : 픻 → ℂ, we consider the analytic functions g : 픻 → ℂ defined by g(z) = (1 + z(f″(z))/f′(z))/(zf′(z)/f(z)) and g(z) = (1 − α)(zf′(z))/f(z) + α(1 + (zf″(z))/f′(z)), 0 ≤ α ≤ 1. We determine the largest radius ρ with 0 < ρ ≤ 1 such that g(ρ z) is subordinate to various functions with positive real part. [ABSTRACT FROM AUTHOR]

The Fekete-Szegö inequality is one of the inequalities for the coefficients which associated with the famous Bieberbach conjecture. Other issues associated with this inequality are determining the Hankel determinant denoted as Hd inequalities which are involved in the study of the singularities and integral coefficients in the Taylor series representations. In this investigation, we discuss such inequalities for certain mappings g for which [Figure: see text](g′(z))α[2zg′(z)g(z)−g(−z)](1−α)≺1+sin⁡z,(0≤α≤1) lies in the image domain starlike about 1 and symmetric about real axis. Particularly, we obtain certain inequality for functions involving Poisson distribution. Furthermore, we also find the second Hankel determinant and other inequalities related to this newly defined class.

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]

Based on the well known notion of the neighborhoods of univalent analytic functions and other related developments, in this article, at first, we obtain an interesting result on neighborhoods of sense-preserving harmonic mappings. Thereafter, we discuss about neighborhoods of stable harmonic univalent mappings and some of its subclasses. [ABSTRACT FROM AUTHOR]

Let H be the class of harmonic functions f = h + g ¯ in the unit disk D : = { z ∈ C : | z | < 1 } , where h and g are analytic in D . In 2020, N. Ghosh and V. Allu introduced the class P H 0 (M) of normalized harmonic mappings defined by P H 0 (M) = { f = h + g ¯ ∈ H : Re (z h ′ ′ (z)) > - M + | z g ′ ′ (z) | with M > 0 , g ′ (0) = 0 , z ∈ D } . In this paper, we investigate various geometric properties such as starlikeness, convexity, convex combination and convolution for functions in the class P H 0 (M) . Furthermore, we determine the sharp Bohr–Rogosinski radius, improved Bohr radius and refined Bohr radius for the class P H 0 (M) . [ABSTRACT FROM AUTHOR]

In this paper, we answer the questions raised in the paper [On the difference of inverse coefficients of univalent functions, Symmetry, 2020, 12(12), art. 2040, 14pp] by Sim and Thomas, and aim to verify the conjecture posed therein in certain cases. For this purpose, we investigate sharp bounds on moduli difference of successive inverse coefficients for certain classes of close-to-convex functions. [ABSTRACT FROM AUTHOR]

Cotîrlă and Szász (Comput Methods Funct Theory: 2023) solved a conjecture related with inclusion relation for Bessel functions, proposed by Baricz and András (Complex Var Elliptic Equ 54(7): 689–696, 2009). In this paper, we prove this conjecture for normalized Struve functions by using subordination factor sequences. [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]

In this paper, we introduce and investigate a subclass of analytic and bi-univalent functions which both f ⁢ (z) and f - 1 ⁢ (z) are m-fold symmetric on the open unit disk. Furthermore, we find upper bounds for the initial coefficients | a m + 1 | and | a 2 ⁢ m + 1 | for functions in this subclass. The results presented in this paper generalize and improve some recent works [ABSTRACT FROM AUTHOR]

If S denotes the class of all univalent functions in the open unit disk D : = z ∈ C : | z | < 1 with the form f (z) = z + ∑ n = 2 ∞ a n z n , then the logarithmic coefficients γ n of f ∈ S are defined by log f (z) z = 2 ∑ n = 1 ∞ γ n (f) z n , z ∈ D. The logarithmic coefficients were brought to the forefront by I.M. Milin in the 1960’s as a method of calculating the coefficients a n for f ∈ S . He concerned himself with logarithmic coefficients and their role in the theory of univalent functions, while in 1965 Bazilevič also pointed out that the logarithmic coefficients are crucial in problems concerning the coefficients of univalent functions. In this paper we estimate the bounds for the logarithmic coefficients | γ n (f) | when f belongs to the class B (α , β) of Bazilevič function of type (α , β) . [ABSTRACT FROM AUTHOR]

Let E X be a unit ball on complex Banach space X and Φ be a convex function such that Φ (0) = 1 and ℜ Φ (ξ) > 0 on D = { z ∈ C : | z | < 1 } . In this paper, we continue the work related to the class Q B Φ (E X) of quasi-convex mappings of type B which have a Φ -parametric representation on E X , where the mappings f ∈ Q B Φ (E X) are k-fold symmetric, k ∈ N. We give the improved Fekete-Szegö inequalities for the class Q B Φ (E X) and establish the sharp bounds of all terms of homogeneous polynomial expansions for some subclasses of Q B Φ (E X) . Our main results are closely related to the Bieberbach conjecture in higher dimensions. [ABSTRACT FROM AUTHOR]

In this existing paper, we examine a connection between certain class of harmonic univalent functions with starlike and convex harmonic functions define in the open unit disk by applying the convolution operator defined by Mittag-Leffler function. Several corollaries and consequences of the main results are also obtained. [ABSTRACT FROM AUTHOR]

Let g 1 and g 2 be any two analytic functions defined in the unit disc which are normalized by the condition g 1 (0) = 1 = g 2 (0) and σ n b - 1 , c (z) be the nth Cesàro mean of type (b - 1 , c) for 1 + b > c > 0 . Then g 1 is generalized Cesàro stable with respect to g 2 , whenever σ n b - 1 , c (g 1 , z) g 1 (z) ≺ 1 g 2 (z) (z ∈ Δ , n ∈ N 0) , where σ n b - 1 , c (g , z) = 1 B n ∑ j = 0 n B n - j b j z j = σ n b - 1 , c (z) ∗ g (z) . The main aim of this article is to prove that (A z + 1) / (B z + 1) η is generalized Cesàro stable with respect to (1 / (B z + 1) η) but not with respect to itself for - 1 ≤ B < A ≤ 0 and 0 < η ≤ 1 . As an application, we obtain new and existing results on Cesàro stability and stability. [ABSTRACT FROM AUTHOR]

We derive general formula for the fourth coefficient of the functions belonging to the Carathéodory class involving the parameters lying in the open unit disk. Further, we obtain sharp upper bounds of initial inverse coefficients for certain close-to-convex functions satisfying any one of the inequalities: Re ((1 - z) f ′ (z)) > 0 , Re ((1 - z 2) f ′ (z)) > 0 , Re ((1 - z + z 2) f ′ (z)) > 0 and Re ((1 - z) 2 f ′ (z)) > 0 . [ABSTRACT FROM AUTHOR]

In this paper, we delve into the fascinating realm of analytic mappings within the open unit disc, focusing on functions that are standardly normalized. Our research is centered on determining the sharp upper bound of the third Hankel determinant for a kth-root transformation applied to a specific subclass of these analytic functions. [ABSTRACT FROM AUTHOR]