## Classes of functions defined by certain differential-integral operators.(English)Zbl 0946.30007

Let $${\mathcal A}(p, k)$$, $$p,k\in\mathbb{N}$$, $$p<k$$, denote the class of functions of the form $$f(z)= z^p+ a_k z^k+ a_{k+1} z^{k+1}+\cdots$$, which are holomorphic in the unit disc $${\mathcal U}$$. Let $$T(\alpha, \beta)$$ denote the class of functions $$f\in{\mathcal A}(p, k)$$ satisfying the following condition: $$\Omega^\alpha_\beta f(z)/z^p\prec(1+ Az)/(1+ Bz)$$, $$z\in{\mathcal U}$$. Denote by $$T_\theta(\alpha,\beta)$$ the subclass of the class $$T(\alpha,\beta)$$ of functions $$f$$ such that $$\arg a_n= \theta$$ for $$a_n\neq 0$$, $$n= k+ 1,k+ 2,\dots$$ . The linear operator $$\Omega^\alpha_\beta:{\mathcal A}(p, k)\to{\mathcal A}(p, k)$$ is defined here by the known operators $$Q^\alpha_\beta$$ and $$\Phi^\alpha_\beta$$ from the papers: J. B. Jung, Y. C. Kim, H. M. Srivastava (1993) and Y. C. Kim, K. S. Lee, H. M. Srivastava (1992). In this paper, the author investigates the class $$T_\theta(\alpha, \beta)$$. Coefficient estimates, distortion theorems, extreme points, the radii of convexity and starlikeness are given.

### MSC:

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

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