Notes on certain analytic functions

Authors: EMEL YAVUZ DUMAN, SHIGEYOSHI OWA

Abstract: Let $\mathcal{A}(n)$ be the class of functions $$f(z)=a_nz^n + a_{n+1}z^{n+1}+\cdots (n\in \mathbb{N}),$$ which are analytic in the open unit disk $\mathbb{U}$, where $a_n \neq 0$. For $f(z)\in \mathcal{A}(n)$, Miller and Mocanu in 1978 showed a very interesting result for $f(z)$. Applying the result due to Miller and Mocanu, we would like to consider some new results for such functions. Our results in this paper are generalizations for results by Nunokawa in 1992.

Keywords: Analytic function, the class $\mathcal{S}(n, \alpha)$, the class $\mathcal{K}(n, \alpha)$

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