Some extremal bounds for subclasses of univalent functions

We study the expected number of zeros of $$P_n(z)=\sum_{k=0}^n\eta_kp_k(z),$$ where $\{\eta_k\}$ are complex-valued i.i.d standard Gaussian random variables, and $\{p_k(z)\}$ are polynomials orthogonal on the unit disk. When $p_k(z)=\sqrt{(k+1)/\pi} z^k$, $k\in \{0,1,\dots, n\}$, we give an explicit formula for the expected number of zeros of $P_n(z)$ in a disk of radius $r\in (0,1)$ centered at the origin. From our formula we establish the limiting value of the expected number of zeros, the expected number of zeros in a radially expanding disk, and show that the expected number of zeros in the unit disk is $2n/3$. Generalizing our basis functions $\{p_k(z)\}$ to be regular in the sense of Ullman--Stahl--Totik, and that the measure of orthogonality associated to polynomials is absolutely continuous with respect to planar Lebesgue measure, we give the limiting value of the expected number of zeros of $P_n(z)$ in a disk of radius $r\in (0,1)$ centered at the origin, and show that asymptotically the expected number of zeros in the unit disk is $2n/3$.

ON SOME CLASSES OF BI-UNIVALENT FUNCTIONS

The investigation of univalent functions is one of the fundamental ideas of Geometric function theory (GFT). However, the class of these functions cannot be investigated as a whole for some particular kind of problems. As a result, the study of its subclasses has been receiving numerous attentions. In this direction, subfamilies of the class of univalent functions that map the open unit disc onto the domain bounded by limaçon of Pascal were recently introduced in the literature. Due to the several applications of this domain in Mathematics, Statistics (hypothesis testing problem) and Engineering (rotary fluid processing machines such as pumps, compressors, motors and engines.), continuous investigation of these classes are of interest in this article. To this end, the family of functions for which $ \frac{\varsigma f^{\prime}(\varsigma)}{f(\varsigma)} $ and $ \frac{(\varsigma f^{\prime}(\varsigma))^{\prime}}{f^{\prime}(\varsigma)} $ map open unit disc onto region bounded by limaçon are studied. Coefficients bounds, Fekete Szeg $ \ddot{ \rm{o}} $ inequalities and the bounds of the third Hankel determinants are derived. Furthermore, the sharp radius for which the classes are linked to each other and to the notable subclasses of univalent functions are found. Finally, the idea of subordination is utilized to obtain some results for functions belonging to these classes.

The main objective of this paper is to introduce and investigate various properties and characteristics of a unified class of analytic functions with negative coefficient in the unit disc. The results presented here involve distortion inequalities and growth and distortion theorem involving fractional integrals and fractional derivatives. Keywords: Analytic functions; Univalent functions; Riemann - Liouville operators ; Growth and distortion theorem; Fractional derivatives.

In this study, a novel integral operator that extends the functionality of some existing integral operators is presented. Specifically, the integral operator acts as the inverse operator to the widely recognized Opoola differential operator. By making use of the integral operator, a certain subclass of analytic univalent functions defined in the unit disk is proposed and investigated. This new class encompasses some familiar subclasses, like the class of starlike and the class of convex functions, while some new ones are introduced. The investigation thereafter covers coefficient inequality, distortion, growth, covering, integral preserving, closure, subordinating factor sequence, and integral means properties. Furthermore, the radii problems associated with this class are successfully addressed. Additionally, a few remarks are provided, to show that the novel integral operator and the new class generalize some existing ones.

Let Ss*(ez) denote the class of starlike functions with respect to symmetric points subordinate to the exponential function, i.e., the functions which satisfy in the unit disk U the condition 2zf′(z)f(z)−f(−z)≺ez(z∈U). We obtained the sharp estimate of the second-order Hankel determinants H2,3(f) and improved the estimate of the third-order H3,1(f) for this functions class Ss*(ez).

This paper examines two subclasses of multivalent analytic functions defined with higher-order derivatives. These classes of functions are generalizations of several known subclasses that have been studied in separate works. Moreover, we find several interesting results for functions in these classes, including subordination results, containment relations, and integral preserving properties.

The main objective of this paper is to establish some sufficient conditions so that a class of normalized Mittag–Leffler-type functions satisfies several geometric properties such as starlikeness, convexity, close-to-convexity, and uniform convexity inside the unit disk. Moreover, pre-starlikeness and k-uniform convexity are discussed for these functions. Some sufficient conditions are also derived so that these functions belong to the Hardy spaces Hp and H∞. Furthermore, the inclusion properties of some modified Mittag–Leffler-type functions are discussed. The various results, which are established in this paper, are presumably new, and their importance is illustrated by several interesting consequences and examples. Some potential directions for analogous further research on the subject of the present investigation are indicated in the concluding section.

For three different normalizations of Bessel functions of first kind, the radius of k-parabolic starlikeness and k-uniform convexity of order $$\alpha $$ are determined. The radius of strong starlikeness and other related radius are also obtained for these functions. We also find optimal parameters for which these functions are k-parabolic starlike and k-uniformly convex in the open unit disk.

Classes of functionsU k, which generalize starlike functions in the same manner that the classV k of functions with boundary rotation bounded bykπ generalizes convex functions, are defined. The radius of univalence and starlikeness is determined. The behavior off α(z) = ∫ 0 z [f'(t)]α dt is determined for various classes of functions. It is shown that the image of |z|<1 underV kfunctions contains the disc of radius 1/k centered at the origin, andV k functions are continuous in |z|≦1 with the exception of at most [k/2+1] points on |z|=1.

It is the purpose of this paper to propose a novel class of convex functions associated with strong η-convexity. A relationship between the newly defined function and an earlier generalized class of convex functions is hereby established. To point out the importance of the new class of functions, some examples are presented. Additionally, the famous Hermite–Hadamard inequality is derived for this generalized family of convex functions. Furthermore, some inequalities and results for strong η-convex function are also derived. We anticipate that this new class of convex functions will open the research door to more investigations in this direction.

By making use of an integral operator due to Noor, a new subclass of analytic functions, denoted by k–$\mathscr{UCV}_n$ (n $\in$ N0 := {0,1,2,...};0 ≤ k < ∞); is introduced. For this class the Fekete-Szego problem is completely settled. The results obtained here also give the Fekete-Szego inequalities for the classes of k-uniformly convex functions and k-parabolic starlike functions.

The purpose of the present paper is to establish connections between various subclasses of analytic univalent functions by applying certain convolution operator involving Poisson distribution series. To be more precise,we investigate such connections with the classes of analytic univalent functions of $$\beta $$ -starlike and $$\beta $$ -uniformly convex functions of order $$\alpha $$ in the open unit disk $${\mathbb {U}}$$ . Further we point out some consequences of our main results.

Consider a family of integral operators and a related family of differential operators, both defined on a class of analytic functions holomorphic in the unit disk, distortion properties of the real part are derived from a general aspect. 1. Integral and differential operators. Let F denote the class of analytic functions which are holomorphic in the unit disk E = {| z |< 1}. Let F nad G be its subclasses consisting of f ∈ F normalized by f(0) = f ′(0) − 1 = 0 and f(0) = 1, respectively. In previous papers [3, 6] we have observed an integral operator L(a) defined on F , which is represented by L(a)f(z) = a ∫

Let $\Cal P$ denote the well-known class of functions of the form $p(z)=1+q_1z+\ldots$ holomorphic in the unit disc $\bold D$ and fulfilling the conditions $Rep(z)>0$ in $\bold D$. Let $0\leq b<1, b<B, 0<\alpha <1$, be fixed real numbers and $zbold F$ a given measurable subset of the unit circle $\bold T$ of Lebesgue measure $2\pi\alpha$. For each $r \in (-\pi,\pi)$, denote by $\bold F_r=\{\xi\in \bold T; e^{-iT}\xi \in \bold F\}$ the set arising by rotation of $\bold F$ through the angle $\tau$. Denote by $\Cal P(B,b,\alpha;\bold F)$ the class of functions $p\in \Cal P$ satisfying the following conditions: there exists $\tau \in (-\pi,\pi)$ such that Rep(^{i\theta})\geq b$ a.e. on $\bold T\ \bold F_r$. In the paper the properties of the class $\Cal P(B,b,\alpha;\bold F)$ for different values of the parameters $B, b, \alpha$ and measurable sets $\bold F$ are examined. This article belongs to the series of papers ([4], [5], [6]) where different classes of functions defined by conditions on the circle $\bold T$ were studied. The results of papers [5], [6] are generalized.