## Convexity and starlikeness of functions defined by a class of integral operators.(English)Zbl 0851.30005

Authors abstract: For $$\Lambda : [0,1] \to \mathbb{R}$$ real-valued monotone decreasing function on $$[0,1]$$ satisfying $$\Lambda (1) = 0$$, $$t \Lambda (t) \to 0$$ as $$t \to 0 +$$ and $$t\Lambda' (t)/(1 - t^2)$$ increasing on $$(0,1)$$, we show that $$M_\Lambda (f) \geq 0$$ for $$f$$ close-to-convex where $M_\Lambda (f) = \inf_{|z |< 1} \int^1_0 \Lambda (t) \left[ \text{Re} f' (zt) - {1 - t \over (1 + t)^3} \right] dt.$ This is analogous to a recent result of R. Fournier and St. Ruscheweyh [Rocky Mt. J. Math. 24, No. 2, 529-538 (1994; Zbl 0818.30013)]. Analogously we obtain the least value of $$\beta$$ so that for $$g$$ analytic in $$|z |< 1$$, $$g(0) = g' (0) - 1 = 0$$, $$\text{Re} [e^{i \alpha} (g' (z) - \beta)] > 0$$, $$\beta < 1$$, the functions $F_1 (z) = z_2 F_1 (1, a; a + b; z)* g(z), \quad 0 < a < 1,\;b > 2$ and $F_2 (z) = {(1 - \alpha) (3 - \alpha) \over 2} \int^1_0 t^{- (\alpha + 1)} (1 - t^2) g(tz) dt, \quad 0 \leq \alpha < 1$ are convex. Here $$_2F_1$$ is the Gaussian hypergeometric function. These results are extended to convexity and order of convexity of convex combinations of the form $$\rho z + (1 - \rho) F(z)$$, $$\rho < 1$$. Corresponding starlikeness results in loc. cit. are also extended to such convex combinations.

### MSC:

 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)

### Keywords:

Gaussian hypergeometric function

Zbl 0818.30013
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