Zayid Abdulhadi

On univalent solutions of the biharmonic equation

We analyze the univalence of the solutions of the biharmonic equation. In particular, we show that if is a biharmonic map in the form,, where is harmonic, then is starlike whenever is starlike. In addition, when,, where and are harmonic, we show that is locally univalent whenever is starlike and is orientation preserving.

On some properties of solutions of the biharmonic equation

In the present paper, the properties of the linear complex operator L(f) = Zfz - Z-fz-, which is defined on the class of complex-valued C1 functions in the plane, are investigated. It is shown that harmonicity and biharmonicity are invariant under the linear operatorL. Results concerning starlikeness and convexity of biharmonic functions versus the corresponding harmonic functions are considered. The operator L can be manipulated to express the conditions in the definitions of starlikeness and convexity in a convenient way.

Univalent Logharmonic Mappings in the Plane

This paper surveys recent advances on univalent logharmonic mappings defined on a simply or multiply connected domain. Topics discussed include mapping theorems, logharmonic automorphisms, univalent logharmonic extensions onto the unit disc or the annulus, univalent logharmonic exterior mappings, and univalent logharmonic ring mappings. Logharmonic polynomials are also discussed, along with several important subclasses of logharmonic mappings.

On the univalence of functions with logharmonic laplacian

In this paper, we prove Rados theorem holds for functions of the form F(z)=r^2L(z),L is logharmonic. We show that if F is of the form F(z)=r^2L(z),|z|<1, where L(z)=h(z)g(z)@? is logharmonic, then F is starlike iff @j(z)=h(z)/g(z) is starlike. In addition, when F(z)=r^2L(z)+H(z),|z|<1, where L is logharmonic and H is harmonic, we give the sufficient conditions for F to be locally univalent.

Univalent logharmonic ring mappings

Univalent logharmonic ring mappings are characterized in terms of univalent starlike mappings. An existence and uniqueness theorem is also given.

The Bohr radius for starlike logharmonic mappings

This paper studies the class consisting of univalent logharmonic mappings in the unit disk , where and are analytic in and is a normalized starlike analytic function. A representation theorem for these mappings is obtained, which yields sharp distortion estimates, and a sharp Bohr radius.

Integral means and arclength inequalities for typically real logharmonic mappings

Abstract In this paper, we use star functions to conclude the integral means inequalities for typically real logharmonic mappings. Moreover, we determine the upper bound for the arclength of typically real logharmonic mappings.

Minimal Surfaces whose Gauss Map Covers Periodically the Pointed Upper Half-Sphere Exactly Once

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On the convex-exponent product of logharmonic mappings

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Dirichlet Problem, Univalency and Schwarz Lemma for Biharmonic Mappings

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