Inclusion theorems of convolution operators associated with normalized hypergeometric functions. (English) Zbl 1104.30008

Assume the notations: \(F(a,b,c; z)\) the Gauss hypergeometric function, \(A\) – the class of functions analytic on the unit disk \(D\) and normalized by the conditions \(f(0)= f'(0)- 1= 0\), \(f*g\) – the Hadamard convolution. For given \(\beta\), \(\gamma\), \(0\leq\gamma< 1\), \(\beta< 1\), \[ P_\gamma(\beta)= \{f\in A:\text{Re}[e^{i\varphi}((1- \gamma)z^{-1} f(z)+\gamma f'(z)- \beta)]\geq 0,\;\varphi\in\mathbb{R},\,z\in D\}. \] The author asked and partially answered the following question. Under what conditions on \(\gamma\), \(f\in P_\gamma(\beta_1)\) and \(f\in P_\gamma(\beta_2)\) implies \(f*g\in P_\gamma(\alpha)\), for some \(\alpha=\alpha(\beta_1,\beta_2)\)? He also, among other results, gave conditions for \(a\), \(b\), \(c\), \(\beta\) and \(\gamma\) under which \(zF(a,b,c; z)\) is in \(P_\gamma(\beta)\) or \(zF(a,b,c;z)* f(z)\in{\mathcal H}^\infty\). Results are given in quite complicated forms. Proofs are highly computational.


30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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[1] Balasubramanian, R.; Ponnusamy, S.; Prabhakaran, D.J., Functional inequalities for the quotients of hypergeometric functions, J. math. anal. appl., 293, 355-373, (2004) · Zbl 1061.30012
[2] R. Balasubramanian, S. Ponnusamy, D.J. Prabhakaran, Convexity of integral transforms and function spaces, Integr. Transf. Spec. Funct., to appear. · Zbl 1109.30009
[3] Balasubramanian, R.; Ponnusamy, S.; Vuorinen, M., On hypergeometric functions and function spaces, J. comput. appl. math., 139, 299-322, (2002) · Zbl 1172.33302
[4] Bateman, H., ()
[5] Carlson, B.C.; Shaffer, D.B., Starlike and prestarlike hypergeometric functions, SIAM J. math. anal., 15, 737-745, (1984) · Zbl 0567.30009
[6] Choi, J.H.; Kim, Y.C.; Saigo, M., Geometric properties of convolution operators defined by Gaussian hypergeometric functions, Integr. transf. spec. funct., 13, 2, 117-130, (2002) · Zbl 1019.30011
[7] Fejér, L., Untersuchunger über potenzreihen mit mehrfach monotoner koeffizientenfolge, Acta litt. sci., 8, 89-115, (1936) · JFM 63.0249.04
[8] Fournier, R.; Ruscheweyh, St., On two extremal problems related to univalent functions, Rocky mountain J. math., 24, 529-538, (1994) · Zbl 0818.30013
[9] Gangadharan, A.; Shanmugam, T.N.; Srivastava, H.M., Generalized hypergeometric functions associated with k-uniformly convex functions, Comput. math. appl., 44, 1515-1526, (2002) · Zbl 1036.33003
[10] Goodman, A.W., Univalent functions, vols. I and II, (1983), Polygonal Publishing House Washington, NJ
[11] Hohlov, Y.E., Convolution operators preserving univalent functions, Ukrain. mat. J., 37, 220-226, (1985), (in Russian)
[12] Y.C. Kim, Some properties of \(\phi\)-fractional integrals, in: Fractional Calculus and Its Applications, International Tokyo Conference Proceedings, College of Engineering, Nihon University, 1990, pp. 80-84, Internat. J. Math. Math. Sci. 22 (4) (1999) 765-773.
[13] Kim, Y.C.; Rønning, F., Integral transforms of certain subclasses of analytic functions, J. math. anal. appl., 258, 466-486, (2001) · Zbl 0982.44001
[14] Kim, Y.C.; Srivastava, H.M., Fractional integral and other linear operators associated with the Gaussian hypergeometric function, Complex variables theory appl., 34, 293-312, (1997) · Zbl 0951.30010
[15] Kiryakova, V.S.; Saigo, M.; Srivastava, H.M., Some criteria for univalence of analytic functions involving generalized fractional calculus operators, Fract. calc. appl. anal., 1, 79-104, (1998) · Zbl 0951.30012
[16] Küstner, R., Mapping properties of hypergeometric functions and convolutions of starlike or convex functions of order \(\alpha\), Comput. meth. funct. theory, 2, 2, 597-610, (2002) · Zbl 1053.30006
[17] Obradović, M.; Ponnusamy, S.; Singh, V.; Vasundhra, P., Starlikeness and convexity applied to certain classes of rational functions, Analysis, 22, 225-242, (2002) · Zbl 1010.30011
[18] Ponnusamy, S., Neighborhoods and Carathéodory functions, J. anal., 4, 41-51, (1996) · Zbl 0867.30009
[19] Ponnusamy, S., Close-to-convexity properties of Gaussian hypergeometric functions, J. comput. appl. math., 88, 327-337, (1997) · Zbl 0901.30007
[20] Ponnusamy, S., Hypergeometric transforms of functions with derivative in a half plane, J. comput. appl. math., 96, 35-49, (1998) · Zbl 0934.30006
[21] Ponnusamy, S.; Rønning, F., Duality for Hadamard products applied to certain integral transforms, Complex variables theory appl., 3, 263-287, (1997) · Zbl 0878.30007
[22] Ponnusamy, S.; Rønning, F., Geometric properties for convolutions of hypergeometric functions and functions with the derivative in a halfplane, Integr. transf. spec. funct., 8, 121-138, (1999) · Zbl 0938.30006
[23] S. Ponnusamy, A. Swaminathan, Convexity of the incomplete beta function, preprint.
[24] Ponnusamy, S.; Vuorinen, M., Asymptotic expansions and inequalities for hypergeometric functions, Mathematika, 44, 278-301, (1997) · Zbl 0897.33001
[25] Ponnusamy, S.; Vuorinen, M., Univalence and convexity properties for Gaussian hypergeometric functions, Rocky mountain J. math., 31, 327-353, (2001) · Zbl 0973.30017
[26] Ruscheweyh, St., Convolutions in geometric function theory, sem. math. sup., vol. 83, (1982), University of Montréal Press
[27] Silverman, H., Starlike and convexity properties for hypergeometric functions, J. math. anal. appl., 172, 574-581, (1993) · Zbl 0774.30015
[28] Swaminathan, A., Certain sufficiency conditions on Gaussian hypergeometric functions, J. ineq. pure appl. math., 5, 4, 1-10, (2004), (Article 83, electronic) · Zbl 1126.30010
[29] Temme, N.M., Special functions. an introduction to the classical functions of mathematical physics, (1996), Wiley New York · Zbl 0863.33002
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