Rosihan M. Ali

Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions ☆

Abstract Estimates on the initial coefficients are obtained for normalized analytic functions f in the open unit disk with f and its inverse g = f − 1 satisfying the conditions that z f ′ ( z ) / f ( z ) and z g ′ ( z ) / g ( z ) are both subordinate to a univalent function whose range is symmetric with respect to the real axis. Several related classes of functions are also considered, and connections to earlier known results are made.

Coefficient bounds for p-valent functions

Abstract Sharp bounds for | a p + 2 - μ a p + 1 2 | and ∣ a p +3 ∣ are derived for certain p -valent analytic functions. These are applied to obtain Fekete-Szego like inequalities for several classes of functions defined by convolution.

Pure 2D picture grammars and languages

A new syntactic model, called pure two-dimensional (2D) context-free grammar (P2DCFG), is introduced based on the notion of pure context-free string grammar. The rectangular picture generative power of this 2D grammar model is investigated. Certain closure properties are obtained. An analogue of this 2D grammar model called pure 2D hexagonal context-free grammar (P2DHCFG) is also considered to generate hexagonal picture arrays on triangular grids.

Subordination and Superordination on Schwarzian Derivatives

Let the functions be analytic and let be analytic univalent in the unit disk. Using the methods of differential subordination and superordination, sufficient conditions involving the Schwarzian derivative of a normalized analytic function are obtained so that either or . As applications, sufficient conditions are determined relating the Schwarzian derivative to the starlikeness or convexity of .

Differential subordination and superordination of analytic functions defined by the Dziok–Srivastava linear operator☆

Abstract Differential subordination and superordination results are obtained for analytic functions in the open unit disk which are associated with the Dziok–Srivastava linear operator. These results are obtained by investigating appropriate classes of admissible functions. Sandwich-type results are also obtained.

Convexity and starlikeness of functions defined by a class of integral operators

For A : [0,1] → R real-valued monotone decreasing function on [0,1] satisfying A(l)=0tA(t)→ 0 as t→0+ and tA′(t)/(l−t2 ) increasing on (0,1), we show that MA(f) ≥ 0 for f close-to-convex where This is analogous to a recent result of Fournier and Ruscheweyh [2]. Analogously we obtain least value of β so that for g analytic in , the functions and are convex. Here 2F1 is the Gaussian hypergeometric function. These results are extended to convexity and order of convexity of convex combinations of the form ρz +(1−ρ)F(z)ρ< 1. Corresponding starlikeness results in [2] are also extended to such convex combinations.

Starlikeness associated with parabolic regions

A parabolic starlike function f of order ρ in the unit disk is characterized by the fact that the quantity zf′(z)/f(z) lies in a given parabolic region in the right half-plane. Denote the class of such functions by PS∗(ρ). This class is contained in the larger class of starlike functions of order ρ. Subordination results for PS∗(ρ) are established, which yield sharp growth, covering, and distortion theorems. Sharp bounds for the first four coefficients are also obtained. There exist different extremal functions for these coefficient problems. Additionally, we obtain a sharp estimate for the Fekete-Szego coefficient functional and investigate convolution properties for PS∗(ρ).

Sufficient Conditions for Janowski Starlikeness

Let A,B,D,E∈[−1,1] and let p(z) be an analytic function defined on the open unit disk, p(0)=1. Conditions on A, B, D, and E are determined so that 1

Subclasses of Meromorphic Functions Associated with Convolution

Several subclasses of meromorphic functions in the unit disk are introduced by means of convolution with a given fixed meromorphic function. Subjecting each convoluted-derived function in the class to be subordinated to a given normalized convex function with positive real part, these subclasses extend the classical subclasses of meromorphic starlikeness, convexity, close-to-convexity, and quasi-convexity. Class relations, as well as inclusion and convolution properties of these subclasses, are investigated.

A Third-Order Differential Equation and Starlikeness of a Double Integral Operator

Functions that are analytic in the unit disk and satisfy the differential equation are considered, where is subordinated to a normalized convex univalent function . These functions are given by a double integral operator of the form with subordinated to . The best dominant to all solutions of the differential equation is obtained. Starlikeness properties and various sharp estimates of these solutions are investigated for particular cases of the convex function .