×

Close-to-convexity properties of Gaussian hypergeometric functions. (English) Zbl 0901.30007

Let \(F(a, b;c;z)\) be the classical hypergeometric function. The sufficient conditions on \(a\), \(b\), \(c\) under which \(zF(a,b; c,z)\) or \({c\over ab} [F(a, b; c;z)- 1]\) is closed-to-convex of order \(\beta\) in \(| z|< 1\) are given.
Example: If \(a\in (0,\infty)\), \(b\in\left(0,{1\over a}\right]\) and if for some real \(\eta\), \(|\eta|< {\pi\over 2}\), \[ \beta\leq 1-{1\over\cos\eta} (1-\Gamma(a+ b)/\Gamma(a)\Gamma(b)) \] then the function \(f(z)= zF(a,b; a+b;z)\) satisfy: \(\text{Re}[e^{i\eta}(1- z)f'(z)- \beta)]>0\), \((| z|< 1)\).

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
33C20 Generalized hypergeometric series, \({}_pF_q\)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] ()
[2] Anderson, G.D.; Barnard, R.W.; Richards, K.C.; Vamanamurthy, M.K.; Vuorinen, M., Inequalities for zero-balanced hypergeometric functions, Trans. amer. math. soc., 347, 1713-1723, (1995) · Zbl 0826.33003
[3] Askey, R.; Askey, R.; Askey, R., S. Ramanujan and hypergeometric and basic hypergeometric series (Russian), S.K. Suslov. uspekhi mat. nauk, S.K. Suslov. uspekhi mat. nauk, Math. surveys, 45, 271, 37-86, (1990), translation in Russian · Zbl 0722.33009
[4] Bateman, H., ()
[5] Duren, P.L., Univalent functions, () · Zbl 0398.30010
[6] Evans, R.J., Ramanujan’s second notebook: asymptotic expansions for hypergeometric series and related functions, (), 537-560
[7] Fournier, R.; Ruscheweyh, St., On two extremal problems related to univalent functions, Rocky mountain J. math., 24, 2, 529-538, (1994) · Zbl 0818.30013
[8] Frideman, B., Two theorems on schlicht functions, Duke math. J., 13, 171-177, (1946)
[9] Hille, E., Hypergeometric functions and conformal mappings, J. differential equations, 34, 147-152, (1979) · Zbl 0387.33004
[10] Krzyż, J., A counterexample concerning univalent functions, Folia societatis scientiarium lubliniensis, Mat. fiz. chem., 2, 57-58, (1962)
[11] Lehto, O.; Virtanen, K.I., Quasiconformal mappings in the plane, () · Zbl 0267.30016
[12] Miller, S.S.; Mocanu, P.T., Univalence of Gaussian and confluent hypergeometric functions, (), 333-342, 2 · Zbl 0707.30012
[13] Nehari, Z., Conformal mapping, (1952), McGraw-Hill New York · Zbl 0048.31503
[14] Ozaki, S., On the theory of multivalent functions, Sci. rep. Tokyo bunrika daigaku sect. A, 2, 167-188, (1935) · JFM 61.0353.02
[15] Ponnusamy, S., Univalence of Alexander transform under new mapping properties, Complex variables theory appl., 30, 1, 55-68, (1996) · Zbl 0865.30007
[16] Ponnusamy, S., Inclusion theorems for convolution product of second order polylogarithms and functions with the derivative in a halfplane, (), 28, Rocky Mountain J. Math., to appear · Zbl 0915.30012
[17] Ponnusamy, S.; Rønning, F., Duality for Hadamard products applied to certain integral transforms, Complex variables theory appl., 32, 263-287, (1997) · Zbl 0878.30007
[18] Ponnusamy, S.; Vuorinen, M., Asymptotic expansions and inequalities for hypergeometric functions, Mathematika, 44, 278-301, (1997) · Zbl 0897.33001
[19] Ponnusamy, S.; Vuorinen, M., Univalence and convexity properties of Gaussian hypergeometric functions, (), 34 · Zbl 0973.30017
[20] Rainville, E.D., Special functions, (1960), Chelsea New York · Zbl 0050.07401
[21] Ruscheweyh, St., Convolution in geometric function theory, (1982), Les Presses de l’Université de Montréal Montréal · Zbl 0575.30008
[22] Ruscheweyh, St.; Singh, V., On the order of starlikeness of hypergeometric functions, J. math. anal. appl., 113, 1-11, (1986) · Zbl 0598.30021
[23] Silerman, H., Starlike and convexity properties for hypergeometric functions, J. math. anal. appl., 172, 574-581, (1993) · Zbl 0774.30015
[24] Whittaker, E.T.; Watson, G.N., A course of modern analysis, (1958), Cambridge University Press Cambridge · Zbl 0108.26903
[25] Zmorovič, V.A., On some problems in the theory of univalent functions (in Russian), (), 83-94
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.