×

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.)

Citations:

Zbl 0818.30013
PDF BibTeX XML Cite
Full Text: DOI