ON A NEW SUBCLASS OF ANALYTIC $P$-VALENT FUNCTIONS

Shigeyoshi Owa

Abstract: There are many classes of analytic and $p$-valent functions in the unit disk U.N.S. Sohi studied a class $S_p(\alpha)$ of analytic and $p$-valent functions $$ f(z)= z^p+ \sum_{n=1}^\infty a_{p+n}z^{p+n},\qquad (p\in N) $$ in the unit disk $U$ satisfying the condition $$ |f'(z)/pz^{p-1}-\alpha|<\alpha,\qquad (z\in U) $$ for $\alpha >1/2$. In this paper, we consider a new subclass $S_{p,k}(\alpha)$ of analytic and $p$-valent functions $$ f(z)= z^p+\sum a_{p+n}z^{p+n},\qquad (p\in N) $$ in the unit disk $U$ satisfying the condition $$ \left|\frac{\Gamma(p+1-k)D^k_z(z)}{\Gamma(p+1)z^{p-k}}\right|<\alpha, \qquad (z\in U) $$ for $01/2$ and $p\in N$, where $D^k_zf(z)$ means the fractional derivative of order $k$ of $f(z)$. It is the purpose of this paper to show a distortion theorem, the coefficient estimates and a convolution theorem for the class $S_{p,k}(\alpha)$. Further we give a theorem about convex set of functions in the class $S_{p,k}(\alpha)$.

Classification (MSC2000): 26A24

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