Abstract
In this paper, we study some properties of analytic functions with fixed initial coefficients. The methodology of differential subordination is used for modification and improvements of several well-known results for subclasses of univalent functions by restricting the functions with fixed initial coefficients. Actually, by extending the Nunokawa lemma for fixed initial coefficient functions, we obtain some novel results on subclasses of univalent functions, such as differential inequalities for univalency or starlikeness of analytic functions. Also, we provide some new sufficient conditions for strongly starlike functions. The results of this paper extend and improve the previously known results by considering functions with fixed second coefficients.
Highlights
Introduction and PreliminariesLet H be the class of analytic functions in the unit disc U = fz : jzj < 1g
We study some properties of analytic functions with fixed initial coefficients
We denote by H β1⁄2a, n the class of analytic functions with a fixed initial coefficient as follows: H β1⁄2a, n
Summary
For a ∈ C and n ∈ N, let us define two wellknown classes of analytic functions as follows: H 1⁄2a, n = f f ∈ H : f ðzÞ = a + anzn+⋯,z ∈ Ug, An We denote by H β1⁄2a, n the class of analytic functions with a fixed initial coefficient as follows: H β1⁄2a, n Ali et al [4] improved the theory of differential subordination by this assumption that the second coefficient of analytic function is fixed. We extend this lemma, and it will be applied to obtain several new results by restricting the functions to have a fixed second coefficient.
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