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A third-order differential equation and starlikeness of a double integral operator. (English) Zbl 1207.30012

Summary: Functions \(f(z)=z+\sum_{n=2}^\infty a_n z^n\) that are analytic in the unit disk and satisfy the differential equation \(f'(z)+\alpha z f''(z)+\beta z^2 f'''(z)=g(z)\) are considered, where \(g\) is subordinated to a normalized convex univalent function \(h\). These functions \(f\) are given by a double integral operator of the form
\[ f(z)=\int_0^1\int_0^1 G(z t^\mu s^\nu) t^{-\mu} s^{-\nu} ds dt \]
with \(G'\) subordinated to \(h\). The best dominant to all solutions of the differential equation is obtained. Starlikeness properties and various sharp estimates of these solutions are investigated for particular cases of the convex function \(h\).

MSC:

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

[1] 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 · doi:10.1080/10652460802045282
[2] R. M. Ali, “On a subclass of starlike functions,” The Rocky Mountain Journal of Mathematics, vol. 24, no. 2, pp. 447-451, 1994. · Zbl 0816.30010 · doi:10.1216/rmjm/1181072410
[3] R. M. Ali and V. Singh, “Convexity and starlikeness of functions defined by a class of integral operators,” Complex Variables. Theory and Application, vol. 26, no. 4, pp. 299-309, 1995. · Zbl 0851.30005
[4] P. N. Chichra, “New subclasses of the class of close-to-convex functions,” Proceedings of the American Mathematical Society, vol. 62, no. 1, pp. 37-43, 1977. · Zbl 0355.30013 · doi:10.2307/2041942
[5] R. Fournier and S. Ruscheweyh, “On two extremal problems related to univalent functions,” The Rocky Mountain Journal of Mathematics, vol. 24, no. 2, pp. 529-538, 1994. · Zbl 0818.30013 · doi:10.1216/rmjm/1181072416
[6] Y. C. Kim and H. M. Srivastava, “Some applications of a differential subordination,” International Journal of Mathematics and Mathematical Sciences, vol. 22, no. 3, pp. 649-654, 1999. · Zbl 0960.30009 · doi:10.1155/S016117129922649X
[7] J. Krzyz, “A counter example concerning univalent functions,” Folia Societatis Scientiarum Lublinensis, vol. 2, pp. 57-58, 1962.
[8] S. Ponnusamy, “Neighborhoods and Carathéodory functions,” Journal of Analysis, vol. 4, pp. 41-51, 1996. · Zbl 0867.30009
[9] H. Silverman, “A class of bounded starlike functions,” International Journal of Mathematics and Mathematical Sciences, vol. 17, no. 2, pp. 249-252, 1994. · Zbl 0796.30006 · doi:10.1155/S0161171294000360
[10] R. Singh and S. Singh, “Starlikeness and convexity of certain integrals,” Annales Universitatis Mariae Curie-Skłodowska. Sectio A. Mathematica, vol. 35, pp. 145-148, 1981. · Zbl 0559.30007
[11] R. Szász, “The sharp version of a criterion for starlikeness related to the operator of Alexander,” Annales Polonici Mathematici, vol. 94, no. 1, pp. 1-14, 2008. · Zbl 1187.30023 · doi:10.4064/ap94-1-1
[12] D.-G. Yang and J.-L. Liu, “On a class of analytic functions with missing coefficients,” Applied Mathematics and Computation, vol. 215, no. 9, pp. 3473-3481, 2010. · Zbl 1182.30025 · doi:10.1016/j.amc.2009.10.043
[13] S. S. Miller and P. T. Mocanu, Differential Subordinations, vol. 225 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2000. · Zbl 0954.34003
[14] S. Ruscheweyh, “New criteria for univalent functions,” Proceedings of the American Mathematical Society, vol. 49, pp. 109-115, 1975. · Zbl 0303.30006 · doi:10.2307/2039801
[15] S. Ruscheweyh and T. Sheil-Small, “Hadamard products of Schlicht functions and the Pólya-Schoenberg conjecture,” Commentarii Mathematici Helvetici, vol. 48, pp. 119-135, 1973. · Zbl 0261.30015 · doi:10.1007/BF02566116
[16] D. J. Hallenbeck and S. Ruscheweyh, “Subordination by convex functions,” Proceedings of the American Mathematical Society, vol. 52, pp. 191-195, 1975. · Zbl 0311.30010 · doi:10.2307/2040127
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