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A note on Ruscheweyh type of integral operators for uniformly $$\alpha$$-convex functions. (English) Zbl 0997.30006
Let $$A$$ denote the class of functions $$f(z)= z+ a_2z^2\cdots$$ analytic in the unit disk $$U$$, $$f\in U$$ is uniformly $$\alpha$$-convex iff $\text{Re}\{(1- \alpha)(z-\zeta) f'(z)/(f(z)- f(\zeta))+ \alpha(1+ (z-\zeta) f''(z))/f'(z))\}> 0$ for all $$z,\zeta\in U$$ and $$0\leq\alpha\leq 1$$. In this note the authors consider the integral operator $F(z)= (F_\alpha(z,\zeta)- F_\alpha(0,\zeta))/F_\alpha'(0,\zeta),$ where $$\alpha> 0$$ and $F_\alpha(z,\zeta)= \Biggl\{(c+ 1/\alpha)/(z- \zeta)^c \int^z_\zeta (t-\zeta)^{c- 1} (f(t)- f(\zeta))^{1/\alpha} dt\Biggr\}^\alpha$ ($$z\in U$$, $$\zeta\in U$$ fixed and $$z=\zeta$$) and prove that $$F$$ is a uniformly $$\alpha$$-convex function when $$f$$ is a uniformly $$\alpha$$-convex function.
##### MSC:
 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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