See Keong Lee

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.

Bounds for the second Hankel determinant of certain univalent functions

The estimates for the second Hankel determinant a2a4−a32 of the analytic function f(z)=z+a2z2+a3z3+⋯ , for which either zf′(z)/f(z) or 1+zf″(z)/f′(z) is subordinate to a certain analytic function, are investigated. The estimates for the Hankel determinant for two other classes are also obtained. In particular, the estimates for the Hankel determinant of strongly starlike, parabolic starlike and lemniscate starlike functions are obtained.MSC:30C45, 30C80.

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 .

Two-dimensional Picture Grammar models

A new theoretical model of grammatical picture generation called extended 2D context-free picture grammar (E2DCFPG) generating rectangular picture arrays of symbols is introduced. This model which allows variables in the grammar and uses the squeezing mechanism of forming the picture language over terminal symbols, is an extension of the pure 2D context-free picture grammar (P2DCFPG)[13]. The extended picture grammar model E2DCFPG is shown to have more picture generative power than the P2DCFPG and certain other existing 2D models. Certain closure and other properties of this new model are also examined.

Convolutions of meromorphic multivalent functions with respect to n-ply points and symmetric conjugate points

Abstract It is well-known that the classes of starlike, convex and close-to-convex univalent functions are closed under convolution with convex functions. In this paper, closure properties under convolution of general classes of meromorphic p-valent functions that are either starlike, convex or close-to-convex with respect to n-ply symmetric, conjugate and symmetric conjugate points are investigated.

A P system model with pure context-free rules for picture array generation

Syntactic models constitute one of the main areas of mathematical studies on picture array generation. A number of 2D grammar models have been proposed motivated by a variety of applications. A P system is a biologically motivated new computing model, proposed by Paun in the area of membrane computing. It is a rich framework for dealing with different problems, including the problem of handling picture array generation. In this paper, the generative power of the array-rewriting P system with pure 2D context-free rules is investigated by comparing it with other 2D grammar models, thus bringing out the suitability of this P system model for picture array generation.

Array P System Model with Pure Context-Free Rules

Motivated by the structure and functioning of the livingcell, a new computing model, called P System, was proposedin the area of Membrane Computing. Among severalareas of application of P systems, one area is concernedwith generation of picture array languages. Ceterchiet al. [1] initially proposed array-rewriting P systemsby extending the notion of string-rewriting P-Systems [8].On the other hand a new theoretical model of grammaticalpicture generation called pure 2D context-free picturegrammar (P2DCFPG) generating rectangular picture arraysof symbols has been introduced by Subramanian et al.[17]. Here we consider P2DCFPG type of rules in arrayrewritingP systems and examine the generating power ofthe resulting array P systems.

Coefficient Conditions for Starlikeness of Nonnegative Order

� ∞ k is starlike of nonnegative order in the unit disk. A result of Vietoris on trigonometric sums is extended in this pursuit. Conditions for close to convexity and convexity in the direction of the imaginary axis are also established. These results are applied to investigate the starlikeness of functions involving the Gaussian hypergeometric functions.

Differential Sandwich Theorem for Multivalent Meromorphic Functions associated with the Liu-Srivastava Operator

Differential subordination and superordination results are obtained for multivalent meromorphic functions associated with the Liu-Srivastava linear operator in the punctured unit disk. These results are derived by investigating appropriate classes of admissible functions. Sandwich-type results are also obtained.

Initial Coefficients of Biunivalent Functions

An analytic function defined on the open unit disk is biunivalent if the function and its inverse are univalent in . Estimates for the initial coefficients of biunivalent functions are investigated when and , respectively, belong to some subclasses of univalent functions. Some earlier results are shown to be special cases of our results.