Abstract
The main object of this investigation is to give some sufficient conditions for analytic functions, by the method of p-subordination chains, to be the pth power of a univalent function in the open unit disk U. Also, the significant relationships and relevance to other results are also given. A number of known univalent conditions would follow upon specializing the parameters involved in our main results. MSC: Primary 30C45; secondary 30C55; 30C80
Highlights
2 p-normalized subordination chain and related theorem Before proving our main theorem, we need a brief summary of the method of psubordination chains
The univalence of complex functions is an important property, but, it is difficult and in many cases impossible to show directly that a certain complex function, especially a function belonging to the class A, is univalent
Deniz et al [ ] submitted a paper which includes sufficient conditions for a integral operator to be the pth power of a univalent function in U
Summary
2 p-normalized subordination chain and related theorem Before proving our main theorem, we need a brief summary of the method of psubordination chains. Let A denote the class of analytic functions in the open unit disk U which satisfy the usual normalization condition f ( ) = f ( ) – = . Let P denote the class of functions p(z) = + Let Ap denote the class of analytic functions in the open unit disk U which satisfy the normalizations f (k)( ) = for k = , , .
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