The projection on the base plane of a regular minimal surface S in R with isothermal parameters defines a complex-valued univalent harmonic function f = h(z) + g(z). The aim of this paper is to obtain the distortion inequalities for the Weierstrass-Enneper parameters of the minimal surface for the harmonic multivalent functions for which analytic part is an m-valent convex function. 2000 Mathem...

which are analytic in the open unit disk U {z : z ∈ C and |z| < 1} and S denote the subclass ofA that are univalent in U. A function f z inA is said to be in class S∗ of starlike functions of order zero in U, if R zf ′ z /f z > 0 for z ∈ U. Let K denote the class of all functions f ∈ A that are convex. Further, f is convex if and only if zf ′ z is star-like. A function f ∈ A is said to be close...

‎In this paper,  the sufficient conditions for the linear combinations of slanted half-plane harmonic mappings to be univalent and convex in the direction of $(-gamma) $ are studied. Our result improves some recent works. Furthermore, a illustrative example and imagine domains of the linear combinations satisfying the desired conditions are enumerated.

Two classes of univalent harmonic functions on unit disc satisfying the conditions ∑∞ n=2(n−α)(|an|+|bn|) ≤ (1−α)(1−|b1|) and ∑∞ n=2 n(n−α)(|an|+|bn|) ≤ (1−α)(1−|b1|) are given. That the ranges of the functions belonging to these two classes are starlike and convex, respectively. Sharp coefficient relations and distortion theorems are given for these functions. Furthermore results concerning th...

Krust established that all conjugate and associate surfaces of a minimal graph over a convex domain are also graphs. Using a convolution theorem from the theory of harmonic univalent mappings, we generalize Krust’s theorem to include the family of convolution surfaces which are generated by taking the Hadamard product or convolution of mappings. Since this convolution involves convex univalent ...

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. 20...

The first author proved that the harmonic convolution of a normalized right half-plane mapping with either another normalized right halfplane mapping or a normalized vertical strip mapping is convex in the direction of the real axis. provided that it is locally univalent. In this paper, we prove that in general the assumption of local univalency cannot be omitted. However, we are able to show t...

Recently, many papers in the theory of univalent functions have been devoted to mapping and characterization properties of various linear integral or integro-differential operators in the class S (of normalized analytic and univalent functions in the open unit disk U), and in its subclasses (as the classes S∗ of the starlike functions and K of the convex functions in U). Among these operators, ...