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Study of the class of univalent functions with negative coefficients defined by Ruscheweyh derivatives. II. (English) Zbl 1138.30015
Let $\Omega$ be the set of all analytic functions in the open unit disk of the form $f(z)=z-\sum^{\infty}_{n=2} a_nz^n$. $f$ in $\Omega$ is said to be in $\Omega (\alpha,\beta,\gamma)$ if Re$\displaystyle\Big\{\frac{z(D^\lambda f(z)^\prime}{(1-\alpha)D^\lambda f(z) +\alpha z^2 (D^\lambda f(z))^{\prime\prime}}\Big\}>\beta$ where $0\le\alpha<1,\ 0\le\beta<1,\ \lambda>-1$ and the operator $D^\lambda f$ is the Ruscheweyh derivative of defined by $\displaystyle D^\lambda f(z)=\frac{z}{(1-z)^{\lambda+1}}*f(z)=z-\sum^{\infty}_{n=2}a_nB_n(\lambda)z^n$ where $\displaystyle B_n(\lambda)=\frac{\Gamma(n+\lambda)}{(n-1)!\Gamma(1+\lambda)}$.This implies $\displaystyle D^\lambda f(z)=\frac{z(z^{\lambda-1} f(z))^{(\lambda)}}{\lambda!}$, $n=0, 1, 2 \dots$. The class $\Omega (\alpha,\beta,\gamma)$ contains many well-known classes, e.g. the starlike functions of order $\beta$ ($\alpha = \lambda =0$) and the convex functions of order $\beta$ ($\alpha = \lambda-1 = 0$). The authors obtain properties for $\Omega (\alpha,\beta,\gamma)$ by using integral operators on the members of the class, e.g. the region of starlikeness and convexity. Fractional calculus techniques are employed to obtain applications for the class. Finally, the authors investigate the neighborhood of a function in $\Omega (\alpha,\beta,\gamma)$ following the concept given by {\it S. Ruscheweyh} [Proc. Am. Math. Soc. 81, (4), 141--145 (1981; Zbl 0458.30008)].
30C45Special classes of univalent and multivalent functions