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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.)

Citations:

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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