## Double integral operators concerning starlike of order $$\beta$$.(English)Zbl 1201.30015

Summary: Double integral operators which were considered by S. S. Miller and P. T. Mocanu [Integral Transforms Spec. Funct. 19, No. 8, 591–597 (2008; Zbl 1156.30014)] are discussed. In order to show the analytic function $$f(z)$$ is starlike of order $$\beta$$ in the open unit disk $$\mathbb U$$, the theory of differential subordinations for analytic functions is applied. The object of the present paper is to discuss some interesting conditions for $$f(z)$$ to be starlike of order $$\beta$$ in $$\mathbb U$$ concerned with second-order differential inequalities and double integral operators.

### MSC:

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

Zbl 1156.30014
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### References:

 [1] D. J. Hallenbeck and St. Ruscheweyh, “Subordination by convex functions,” Proceedings of the American Mathematical Society, vol. 52, pp. 191-195, 1975. · Zbl 0311.30010 [2] H. Al-Amiri and P. T. Mocanu, “Some simple criteria of starlikeness and convexity for meromorphic functions,” Mathematica, vol. 37(60), no. 1-2, pp. 11-20, 1995. · Zbl 0884.30009 [3] S. S. Miller and P. T. Mocanu, “Double integral starlike operators,” Integral Transforms and Special Functions, vol. 19, no. 7-8, pp. 591-597, 2008. · Zbl 1156.30014 [4] M. Obradović, “Simple sufficient conditions for univalence,” Matematichki Vesnik, vol. 49, no. 3-4, pp. 241-244, 1997. · Zbl 0992.30005
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