A GENERALIZATION OF CERTAIN CLASS OF ANALYTIC FUNCTIONS WITH NEGATIVE COEFFICIENTS

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M. K. AOUF
A. SHAMANDY

Abstract




We introduce the subclass $T^*(A,B,n,a)$ ($-1 \le A < B\le 1$, $0 < B \le 1$, $n \ge 0$, and $0\le\alpha <1$) of analytic func;tions with negative coefficients by the operator $D^n$. Coefficient estimates, distortion theorems, closure theorems and radii of close-to-convexety, starlikeness and convexity for the class $T^*(A,B,n,a)$ are determined. We also prove results involving the modified Hadamard product of two functions associated with the class $T^*(A,B,n,a)$. Also we obtain Several interesting distortion theorems for certain fractional operators .of functions in the class $T^*(A,B,n,a)$. Also we obtain class perserving integral operator of the form


\[F(z)=\afrc{c+1}{z^c}\int_0^z t^{c-1}f(t) dt, \quad c>-1\]





for the class $T^*(A,B,n,a)$. Conversely when $F(z) \in T*(A,B,n,a)$, radius of univalence of $f(z)$ defined by the above equation is obtained.







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How to Cite
AOUF, M. K., & SHAMANDY, A. (1995). A GENERALIZATION OF CERTAIN CLASS OF ANALYTIC FUNCTIONS WITH NEGATIVE COEFFICIENTS. Tamkang Journal of Mathematics, 26(2), 107–117. https://doi.org/10.5556/j.tkjm.26.1995.4384
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