On the Radius of Univalence of Certain Analytic Functions

Abstract:We use Ruscheweyh derivative to define certain new classes of analytic functions with bounded radius rotation and related to conic domains. Some interesting and significant results such as inclusion results, growth rate of coefficients and radius problems for these new classes of k-uniformly functions. Several special cases are discussed. Results obtained in this paper may stimulate further research activities in this field.

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“…The operator J c was studied earlier by Libera [20] and Livingston [21]. Now, we define the Bernardi integral operator J c (f ) on the class S H of harmonic univalent functions of the form (1.1) as follows: …”

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Some properties of generalized convolution of harmonic univalent functions

Porwal

Dixit

2013

Demonstratio Mathematica

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Abstract. The purpose of the present paper is to investigate some interesting properties on generalized convolutions of functions for the classes HP * (α), HS(α) and HC(α). Further, an application of the convolution on certain integral operator are mentioned. IntroductionA continuous complex-valued function f = u + iv defined in a simply connected complex domain D is said to be harmonic in D, if both u and v are real harmonic in D. In any simply connected domain we can write f = h+g, where h and g are analytic in D. We call h the analytic part and g the coanalytic part of f. A necessary and sufficient condition for f to be locally univalent and sense-preserving in D is that |h ′ (z)| > |g ′ (z)|, z ∈ D, (see ). For more basic results on harmonic functions one may refer following standard introductory text book by Duren [14], (see also Ahuja [1] and Ponnusamy and Rasila [28,29]Denote by S H the class of functions f = h+g that are harmonic univalent and sense-preserving in the unit disc U = {z : |z| < 1} for which f (0) = f z (0) − 1 = 0. Then, for f = h + g ∈ S H , we may express the analytic functions h and g asA function f (z) of the form (1.1) in S H is said to be harmonic starlike of order α, (0 ≤ α < 1) in U , if and only if (1.2) ∂ ∂θ {arg f (z)} > α, z ∈ U,2000 Mathematics Subject Classification: 30C45, 26D15.

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On the Radius of Univalence of Certain Analytic Functions

Abstract: It is possible that these together with the constant unitary matrices generate the whole class of such functions, but we have not been able to prove it.

Cited by 42 publications

References 1 publication

On quasi‐convex functions and related topics

On quasi‐convex functions and related topics

Some Classes of k-Uniformly Functions with Bounded Radius Rotation

Some properties of generalized convolution of harmonic univalent functions

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