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In the present work, the authors introduce a new linear differential operator associated with $ q $-calculus which uses the $ q $-derivative operator. For the new class involving the linear differential operator, interesting properties like coefficient inequality, growth, distortion, convolution and radii of starlikeness are established. Various connections of the present class with that of the earlier works are pointed out. [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]

Differential subordination in the complex plane is the generalization of a differential inequality on the real line. In this paper, we consider two subclasses of univalent functions associated with the trigonometric function cos z. Using some properties of the hypergeometric functions, we determine the sharp estimate on the parameter β such that the analytic function p(z) satisfying p(0) = 1, is subordinate to cos z when the differential expression p(z) + βz(dp(z)/dz) is subordinate to the Janowski function. We compute sharp bounds on coefficient functional Hermitian-Toeplitz determinants of the third and the fourth order with an invariance property for such functions. In addition, we estimate bound on Hankel determinants of the second and the third order. [ABSTRACT FROM AUTHOR]

In this paper, we defined a new family of meromorphic functions whose analytic characterization was motivated by the definition of the multiplicative derivative. Replacing the ordinary derivative with a multiplicative derivative in the subclass of starlike meromorphic functions made the class redundant; thus, major deviation or adaptation was required in defining a class of meromorphic functions influenced by the multiplicative derivative. In addition, we redefined the subclass of meromorphic functions analogous to the class of the functions with respect to symmetric points. Initial coefficient estimates and Fekete–Szegö inequalities were obtained for the defined function classes. Some examples along with graphs have been used to establish the inclusion and closure properties. [ABSTRACT FROM AUTHOR]

In this article, we make use of convex analytic functions Ha(z) = [1/(1 - a)] log[(1 - az)/(1 - z)], a - ℝ, |a| ≤ 1, a ≠ 1 and starlike analytic functions Lb(z) = z/[(1 - bz)(1 - z)], b - ℝ, |b| ≤ 1, to construct univalent harmonic functions by means of a transformation on some normalized univalent analytic functions. Besides exploring mapping properties of harmonic functions so constructed, we establish sufficient conditions for their harmonic convolutions or Hadamard products to be locally univalent and sense preserving, univalent and convex in some direction. [ABSTRACT FROM AUTHOR]

In this paper, our aim is to determine the radii of starlikeness and convexity associated with lemniscate of Bernoulli for three different kinds of normalizations of the function Nv (z) = az²J"v (z)+bzJ'v(z)+cJv (z), where Jv is the Bessel function of the first kind of order v. The key tools in the proof of our main results are the Mittag-Leffler expansion for the function Nv (z) and properties of real zeros of it. Also, we give tables related with special cases of parameters. [ABSTRACT FROM AUTHOR]

This article presents new classes of ζ -uniformly q-starlike and qconvex 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]

In the current paper, we introduce new subclasses of analytic and bi-univalent functions involving Caputo-type fractional derivatives subordinating to the Lucas polynomial. Furthermore, we find non-sharp estimates on the first two Taylor–Maclaurin coefficients a 2 and a 3 for functions in these subclasses. Using the values of a 2 and a 3 , we determined Fekete–Szegő inequality for functions in these subclasses. Moreover, we pointed out some more subclasses by fixing the parameters involved in Lucas polynomial and stated the estimates and Fekete–Szegő inequality results without proof. [ABSTRACT FROM AUTHOR]

In the present paper, we propose new generalized classes of (p,q)-starlike and (p,q)-convex functions. These classes are introduced by making use of a (p,q)-derivative operator. There are established Fekete–Szegö estimates | a 3 − μ a 2 2 | for functions belonging to the newly introduced subclasses. Certain subclasses of analytic univalent functions associated with quasi-subordination are defined. [ABSTRACT FROM AUTHOR]

The main purpose of this work is to study the third Hankel determinant for classes of Bernoulli lemniscate-related functions by introducing new subclasses of star-like functions represented by S L λ * and R L λ . In many geometric and physical applications of complex analysis, estimating sharp bounds for problems involving the coefficients of univalent functions is very important because these coefficients describe the fundamental properties of conformal maps. In the present study, we defined sharp bounds for function-coefficient problems belonging to the family of S L λ * and R L λ . Most of the computed bounds are sharp. This study will encourage further research on the sharp bounds of analytical functions related to new image domains. [ABSTRACT FROM AUTHOR]

In the present paper, we study the shifted hypergeometric function f(z)=z_{2}F_{1}(a,b;c;z) for real parameters with 0

In this paper, we derive several fuzzy differential subordination and fuzzy differential superordination results for analytic functions Ms,γξ,β, which involve the extended Mittag-Leffler function and the Pascal distribution series. We also investigate and introduce a class MBF,s,γξ,β (ρ) of analytic and univalent functions in the open unit disc D by employing the newly defined operator Ms,γξ,β. We determine a specific relationship of inclusion for this class. Further, we establish prerequisites for a function role in serving as both the fuzzy dominant and fuzzy subordinant of the fuzzy differential subordination and superordination, respectively. Some novel results that are sandwich-type can be found here. [ABSTRACT FROM AUTHOR]

In the recent years, the study of the Hankel determinant problems have been widely investigated by many researchers. We were essentially motivated by the recent research going on with the Hankel determinant and other coefficient bounds problems. In this research article, we first considered the subclass of analytic starlike functions connected with the domain of the tangent function. We then derived the initial four sharp coeffcient bounds, the sharp Fekete-Szegö inequality, and the sharp second and third order Hankel determinant for the defined class. Also, we derived sharp estimates like sharp coefficient bounds, Fekete-Szegö estimate, and sharp second order Hankel determinant for the functions having logarithmic coefficient and for the inverse coefficient, respectively, for the defined functions class. [ABSTRACT FROM AUTHOR]

This article uses the Mittag-Leffler transformation to explore a specific category of harmonic meromorphic functions. The Mittag-Leffler transformation is a crucial tool for analysing meromorphic functions and provides essential properties and insights into their behavior. The main focus of this study is on harmonic meromorphic functions that can be represented by the MittagLeffler transformation. Furthermore, this research introduces an innovative derivative operator that incorporates this transformation into the domain of harmonic meromorphic functions. The Mittag-Leffler transformation is widely recognised as a powerful technique for analysing various mathematical functions, especially those with fractional order derivatives. It improves our understanding of harmonic meromorphic functions and their inherent characteristics. The research findings highlight the effectiveness of this new derivative operator in unraveling the complexities of these functions. They provide valuable insights into their behavior and fundamental traits. Additionally, the study offers coefficient inequalities, the distortion theorem, distortion bounds, extreme points, convex combinations, and convolution analyses specifically tailored to functions within this particular class. [ABSTRACT FROM AUTHOR]

Let S cos * denote the class of normalized analytic functions f in the open unit disk D satisfying the subordination z f ′ (z) f (z) ≺ cos z . In the first result of this article, we find the sharp upper bounds for the initial coefficients a 3 , a 4 and a 5 and the sharp upper bound for module of the Hankel determinant | H 2 , 3 (f) | for the functions from the class S cos * . The next section deals with the sharp upper bounds of the logarithmic coefficients γ 3 and γ 4 . Then, in addition, we found the sharp upper bound for H 2 , 2 F f / 2 . To obtain these results we utilized the very useful and appropriate Lemma 2.4 of N.E. Cho et al., which gave a most accurate description for the first five coefficients of the functions from the Carathéodory's functions class, and provided a technique for finding the maximum value of a three-variable function on a closed cuboid. All the maximum found values were checked by using MAPLE™ computer software, and we also found the extremal functions in each case. All of our most recent results are the best ones and give sharp versions of those recently published by Hacet. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study certain geometric properties of a class of analytic functions involving the Barnes–Mittag-Leffler function. The main mathematical tools are the monotonicity patterns of some class of functions associated with the gamma and digamma functions. Furthermore, some consequences and examples are presented. [ABSTRACT FROM AUTHOR]

In the present paper, we are merging two interesting and well-known classes, namely those of Bazilevič and close-to-convex functions associated with a new derivative operator. We derive coefficient estimates for this broad category of analytic, univalent and bi-univalent functions and draw attention to the Fekete–Szegö inequalities relevant to functions defined within the open unit disk. Additionally, we identify several specific special cases of our results by specializing the parameters. [ABSTRACT FROM AUTHOR]

The paper mainly investigates the initial coefficients for the subclasses of starlike functions defined by using the Sine function involving a (0 = a < 1). We obtain upper bounds for the second and third Toeplitz determinants whose elements are the initial coefficients. Also, we obtain upper bounds of initial logarithmic coefficients for the subclass of starlike functions of order a which are connected with the sine function. Parts of the results we obtained generalized and improved previous work. [ABSTRACT FROM AUTHOR]

In this paper, we shall find the order of starlikeness and convexity for integral operators Fαj,βj,λj,ζ (z) = {ζ ∫0z tζ−1 πnj=1 (Eαj,βj(t)/t )1/λj dt}1/ζ, where the functions Eαj,βj are the normalized Mittag-Leffler functions. [ABSTRACT FROM AUTHOR]

Let u$u$ be a subharmonic function on the open unit disc D$\mathbb {D}$, centered at the origin of the complex plane, and let φ:D→D$\varphi:\mathbb {D}\rightarrow \mathbb {D}$ be a holomorphic function such that φ(0)=0$\varphi (0)=0$. A classical result, known as Littlewood subordination principle, states Iu∘φ(r)⩽Iu(r)$\mathrm{I}_{u\circ \varphi }(r)\leqslant \mathrm{I}_u(r)$, where Iu∘φ(r)$\mathrm{I}_{u\circ \varphi }(r)$ and Iu(r)$\mathrm{I}_u(r)$ are integral means over the circle of radius 0⩽r<1$0\leqslant r<1$ centered at the origin, of the functions u∘φ$u\circ \varphi$ and u$u$, respectively. In this note, we obtain an unexpected improvement of Littlewood subordination principle in the case when the function φ$\varphi$ is univalent, by proving that Iu∘φ(r)⩽Iu(r)−κ∫0rIu(r)−Iu(t)dt,$$\begin{equation*} \hspace*{40pt}\mathrm{I}_{u\circ \varphi }(r)\leqslant \mathrm{I}_u(r)-\kappa \int _0^{r}{\left(\mathrm{I}_u(r)-\mathrm{I}_u(t)\right)}\mathrm{d}t, \end{equation*}$$where κ=(1−|φ′(0)|)/2$\kappa =(1-|\varphi ^{\prime }(0)|)/2$. We also list some applications of this result including an improved variant of Rogosinski theorem with univalent symbol. [ABSTRACT FROM AUTHOR]

In this work, we study some new applications of symmetric quantum calculus in the field of Geometric Function Theory. We use the cardioid domain and the symmetric quantum difference operator to generate new classes of multivalent q-starlike and q-convex functions. We examine a wide range of interesting properties for functions that can be classified into these newly defined classes, such as estimates for the bounds for the first two coefficients, Fekete–Szego-type functional and coefficient inequalities. All the results found in this research are sharp. A number of well-known corollaries are additionally taken into consideration to show how the findings of this research relate to those of earlier studies. [ABSTRACT FROM AUTHOR]

The concept of subordination is expanded in this study from the fuzzy sets theory to the geometry theory of analytic functions with a single complex variable. This work aims to clarify fuzzy subordination as a notion and demonstrate its primary attributes. With this work's assistance, new fuzzy differential subordinations will be presented. The first theorems lead to intriguing corollaries for specific aspects chosen to exhibit fuzzy best dominance. The work introduces a new integral operator for meromorphic functions and uses the newly developed integral operator, which is starlike and convex, respectively, to obtain conclusions on fuzzy differential subordination. [ABSTRACT FROM AUTHOR]

On the class of typically real odd polynomials of degree 2 N - 1 F (z) = z + ∑ j = 2 N a j z 2 j - 1 we consider two problems: 1) stretching the central unit disc under the above polynomial mappings and 2) estimating the coefficient a 2. It is shown that | F (z) | ≤ 1 2 csc 2 π 2 N + 2 , - 1 + 4 sin 2 π 2 N + 4 ≤ a 2 ≤ - 1 + 4 cos 2 π N + 2 for odd N , and - 1 + 4 (ν N) 2 ≤ a 2 ≤ - 1 + 4 cos 2 π N + 2 for even N , where ν N is a minimal positive root of the equation U N + 1 ′ (x) = 0 with U N + 1 ′ (x) being the derivative of the Chebyshev polynomial of the second kind of the corresponding order. The above boundaries are sharp, the corresponding estremizers are unique and the coefficients are determined. [ABSTRACT FROM AUTHOR]

The first aim is to prove that the Roper-Suffridge extension operator preserves ε -starlike property on general domains given by convex functions. The second is to construct the generalized Roper-Suffridge extension operator on Reinhard domains Ω p 1 , p 2 , ⋯ , p n = { (z 1 , ... , z n) ∈ C n : ∑ j = 1 n | z j | p j < 1 } , p 1 , ⋯ , p n ≥ 1. By using a refined Schwarz-Pick lemma, we prove that the operator preserves important properties, e.g., subordination property and spirallikeness. This solves a problem of Gong and Liu. Our result improves many known results from p 1 = 2 to 1 ≤ p 1 < ∞ . As applications, we obtain growth and covering results associated with the extension operator on p-unit ball. Finally, by obtaining geometric and analytic properties of bounded symmetric domains, we generalize the Pfaltzgraff-Suffridge extension operator over bounded symmetric domains and prove Loewner chains and starlikeness are also preserved with a new idea. Further, we propose two conjectures for convexity property. [ABSTRACT FROM AUTHOR]

The present extensive study is focused to find estimates for the upper bounds of the Toeplitz determinants. The logarithmic coefficients of univalent functions play an important role in different estimates in the theory of univalent functions, and in the this paper we derive the estimates of Toeplitz determinants and Toeplitz determinants of the logarithmic coefficients for the subclasses LsSqp, LsC qp, and LsSqp ∩ S, LsC q p ∩ S, 0 < q ≤ p ≤ 1, respectively, defined by post quantum operators, which map the open unit disc D onto the domain bounded by the lima¸con curve defined by ∂Ds := { u + iv ∈ C : (u − 1)² + v 2 − s4 } = 4s² (u − 1 + s² ) ² + v² o, where s ∈ [−1, 1] \ {0}. [ABSTRACT FROM AUTHOR]

This work is an attempt to apply Lambert series in the theory of univalent functions. We first consider the Hadamard product of Rabotnov function and Lambert series with coefficients derived from the arithmetic function σ n to introduce a normalized linear operator J R α , β z . We then acquire sufficient conditions for J R α , β z to be univalent, starlike and convex, respectively. Furthermore, we discuss the inclusion results in some special classes, namely, spiral-like and convex spiral-like subclasses. In addition, we extend the findings by incorporating two Robin's inequalities, one of which is analogous to the Riemann hypothesis. [ABSTRACT FROM AUTHOR]

In this paper, by using Poisson distribution probability, some characteristics of analytic multivalent q -symmetric starlike and q -symmetric convex functions of order η are examined. Then, by utilizing the Poisson distribution and the concept of the q -analogue Salagean integral operator, the p -valent convergence polynomial was introduced. Furthermore, a number of subclasses of analytic symmetric p -valent functions linked to novel polynomials are also deduced. After that, specific coefficient constraints are determined and symmetric δ , q -neighborhoods for p -valent functions are defined. In relation to symmetric δ , q -neighborhoods of q -symmetric p -valent functions formed by Poisson distributions, this paper presents new inclusion results. In addition, a detailed discussion of certain q -symmetric inequalities of analytic functions with negative coefficients is also provided. [ABSTRACT FROM AUTHOR]

In the present paper, we consider the class A ¯ of functions f (z) of the form f (z) = z + ∑ k = 1 ∞ a 1 + k 3 z 1 + k 3 that are analytic in the open unit disc U. If a 1 + k 3 = 0 for k ≠ 3 n   (n = 1 , 2 , 3 , ⋯) , then f (z) is given by f (z) = z + ∑ k = 2 ∞ a k z k. For such functions f (z) ∈ A ¯ , some interesting properties for subordinations and strongly starlike functions are given. Also, some interesting examples for the results are shown. [ABSTRACT FROM AUTHOR]

The object of the present paper is to derive some argument properties associated with meromorphic close-to-starlike and Bazilević functions in the punctured open unit disk. Furthermore, we investigate integral-preserving properties in a sector with some special cases of the main results presented here. [ABSTRACT FROM AUTHOR]

In this paper, we give upper bounds of the Hankel determinants H2,1(f) and H2,2(f) for the classes S*(λ,n), where f is analytic in the open unit disk Δ = {z ∈ C: |z| < 1g and normalized so that f(z) = z+a2z²+a3z³+... [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]

The aim of this paper is to define the linear operator based on the generalized Mittag-Leffler function and the Lambert series. By using this operator, we introduce a new subclass of β-uniformly starlike functions Τ J (α i) . Further, we obtain coefficient estimates, convex linear combinations, and radii of close-to-convexity, starlikeness, and convexity for functions f ∈ Τ J (α i) . In addition, we investigate the inclusion conditions of the Hadamard product and the integral transform. Finally, we determine the second Hankel inequality for functions belonging to this subclass. [ABSTRACT FROM AUTHOR]

In this study, using the ideas of subordination and the quantum-difference operator, we established a new subclass S*(δ,σ, q) of q-starlike functions and the subclass S*σ (δ,σ, q) of q-starlike bi-univalent functions associated with the vertical strip domain. We examined sharp bounds for the first two Taylor-Maclaurin coefficients, sharp Fekete-Szegö type problems, and coefficient inequalities for the function h that belong to S*(δ,σ, q), as well as sharp bounds for the inverse function h that belong to S*(δ,σ, q). We also investigated some results for the class of bi-univalent functions S*σ (δ,σ, q) and well-known corollaries were also highlighted to show connections between previous results and the findings of this paper. [ABSTRACT FROM AUTHOR]

This paper considers a special class S L (h) of logharmonic mappings of the form f (z) = h (z) h ′ (z) ¯ , where h is analytic in the unit disk U, normalized by h (0) = 0 , h ′ (0) = 1 , and h(U) is starlike. For this class of functions, a distortion theorem is proved, and Bohr's inequality along with some improvements and refinements is investigated. In addition, the radius of starlikeness and an estimate for arclength are obtained. [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]

Let $\mathcal {K}_u$ denote the class of all analytic functions f in the unit disk $\mathbb {D}:=\{z\in \mathbb {C}:|z| , normalised by $f(0)=f'(0)-1=0$ and satisfying $|zf'(z)/g(z)-1| in $\mathbb {D}$ for some starlike function g. Allu, Sokól and Thomas ['On a close-to-convex analogue of certain starlike functions', Bull. Aust. Math. Soc. 108 (2020), 268–281] obtained a partial solution for the Fekete–Szegö problem and initial coefficient estimates for functions in $\mathcal {K}_u$ , and posed a conjecture in this regard. We prove this conjecture regarding the sharp estimates of coefficients and solve the Fekete–Szegö problem completely for functions in the class $\mathcal {K}_u$. [ABSTRACT FROM AUTHOR]

We present a canonical series expansion for generalized harmonic functions in the open unit disc in the complex plane that generalizes that recently obtained for the class of $ (p,q) $ (p , q) -harmonic functions. [ABSTRACT FROM AUTHOR]

The focus of the present work is on the establishment and investigation of the coefficient estimates of two new subclasses of bi-close-to-convex functions and bi-concave functions; these are of an Ozaki type and involve a modified Caputo's fractional operator that is associated with three-leaf functions in the open unit disc. The classes are defined using the notion of subordination based on the previously established fractional integral operators and classes of starlike functions associated with a three-leaf function. For functions in these classes, the Fekete-Szegö inequalities and the initial coefficients, | a 2 | and | a 3 | , are discussed. Several new implications of the findings are also highlighted as corollaries. [ABSTRACT FROM AUTHOR]

Without qualms, studies show that quantum calculus has received great attention in recent times. This can be attributed to its wide range of applications in many science areas. In this exploration, we study a new qdifferential operator that generalized many known differential operators. The new q-operator and the concept of subordination were afterwards, used to define a new subclass of analytic-univalent functions that invariably consists of several known and new generalizations of starlike functions. Consequently, some geometric properties of the new class were investigated. The properties include coefficient inequality, growth, distortion and covering properties. In fact, we solved some radii problems for the class and also established its subordinating factor sequence property. Indeed, varying some of the involving parameters in our results led to some existing results. [ABSTRACT FROM AUTHOR]

In this paper, we expand on the notion of the q-derivative (or q-difference) operator for meromorphic multivalent functions, define the higher-order q-derivative operator for meromorphic multivalent functions associated with quantum calculus, and introduce new subclasses of meromorphic multivalent q-starlike functions in connection with Janowski functions. We investigate a number of useful properties of the Janowski functions and higher-order q-derivative operator for a new class of meromorphic multivalent q-starlike functions. Among the many potential uses of this class that we investigate are coefficient estimates, distortion theorems, partial sums, the radius of starlikeness, and a few other well-established results. [ABSTRACT FROM AUTHOR]

We introduce a class R α q of q-prestarlike functions of order α by using q-difference operator. We obtain necessary and sufficient conditions involving convolution for functions to be in the class R α q. We prove that the well-known class of analytic prestarlike functions, R α , is properly contained in R α q. Apart from finding bounds on some initial coefficients of functions in the class R α q , we also investigate some convolution properties of functions in the class R α q. The results of the present manuscript essentially generalize some well-known results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, considering the various important applications of Miller–Ross functions in the fields of applied sciences, we introduced a new class of analytic functions f , utilizing the concept of Miller–Ross functions in the region of the Janowski domain. Furthermore, we obtained initial coefficients of Taylor series expansion of f , coefficient inequalities for f − 1 and the Fekete–Szegö problem. We also covered some key geometric properties for functions f in this newly formed class, such as the necessary and sufficient condition, convex combination, sequential subordination and partial sum findings. [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]

We investigate the sharp bound of certain coefficient functionals associated with a Hankel determinant of the second kind for the inverse function when f belongs to the class of starlike functions with respect to symmetric points. [ABSTRACT FROM AUTHOR]

This paper introduces fractional operators in the complex domain as generalizations for the Srivastava–Owa operators. Some properties for the above operators are also provided. We discuss the convexity and starlikeness of the generalized Libera integral operator. A condition for the convexity and starlikeness of the solutions of fractional differential equations is provided. Finally, a fractional differential equation is converted into an ordinary differential equation by wave transformation; illustrative examples are provided to clarify the solution within the complex domain. [ABSTRACT FROM AUTHOR]

The article aims to estimate the coefficient bounds for the second Hankel determinant by using the class of bi-close-to-convex functions of a complex order in the open unit disk. Making the direct application of Carathéodory function along with the closely related properties of starlike functions, we obtain the upper bound for the second Hankel determinant via certain subclass of bi-close-to-convex functions of complex order. The study discusses the maximization of the second Hankel determinant in both conventional graph and analytic methods. Moreover, we explore and modify some results on the study of bi-close-to-convex functions and its second degree Hankel determinant. At the end of the article, we remark on improvement in the earlier work of some researchers and discover a better value than the one they obtained. [ABSTRACT FROM AUTHOR]

In the present paper, we investigated some conditions to be in the class of Carathéodory functions by using the concept of the first-order differential subordinations. Moreover, various interesting special cases were considered in the geometric function theory as applications of main results presented here. [ABSTRACT FROM AUTHOR]

This paper revealed new fractional calculus applications of special functions in the geometric function theory. The aim of the study presented here was to introduce and begin the investigations on a new fractional calculus integral operator defined as the fractional integral of order λ for the Bessel function of the first kind. The focus of this research was on obtaining certain geometric properties that give necessary and sufficient univalence conditions for the new fractional calculus operator using the methods associated to differential subordination theory, also referred to as admissible functions theory, developed by Sanford S. Miller and Petru T. Mocanu. The paper discussed, in the proved theorems and corollaries, conditions that the fractional integral of the Bessel function of the first kind must comply in order to be a part of the sets of starlike functions, positive and negative order starlike functions, convex functions, positive and negative order convex functions, and close-to-convex functions, respectively. The geometric properties proved for the fractional integral of the Bessel function of the first kind recommend this function as a useful tool for future developments, both in geometric function theory in general, as well as in differential subordination and superordination theories in particular. [ABSTRACT FROM AUTHOR]

In this article, we define a generalized q-integral operator on multivalent functions. It generalizes many known linear operators in Geometric Function Theory (GFT). Inclusions results, convolution properties and q-Bernardi integral preservation of the subclasses of analytic functions are discussed. [ABSTRACT FROM AUTHOR]