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Hypergeometric transforms of functions with derivative in a half plane. (English) Zbl 0934.30006

The author proves inequalities for the Taylor coefficients of the Gaussian hypergeometric function and of the confluent hypergeometric function. These inequalities enable him to get certain geometric properties of the images of the unit disk under such transformations. These geometric properties are used to prove invariance properties of operators defined by convolution with hypergeometric functions.

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
33C15 Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\)
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[1] Aksent’ev, L.A.; Avkhadiev, F.G., The main results on sufficient conditions for an analytic function to be schlicht, Russian math. surveys, 30, 4, 1-64, (1975) · Zbl 0331.30016
[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] Bateman, H., ()
[4] Bernardi, S.D., Convex and starlike univalent functions, Trans. amer. math. soc., 135, 429-446, (1969) · Zbl 0172.09703
[5] de Branges, L., A proof of the Bieberbach conjecture, Acta math., 154, 137-152, (1985) · Zbl 0573.30014
[6] Duren, P.L., ()
[7] Fejér, L., Untersuchungen über potenzreihen mit mehrfach monotoner koeffizientenfolge, Acta litt. scientiarum, 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, 2, 529-538, (1994) · Zbl 0818.30013
[9] Goodman, A.W., ()
[10] Hille, E., Hypergeometric functions and conformal mappings, J. differential equations, 34, 147-152, (1979) · Zbl 0387.33004
[11] Krzyż, J., A counterexample concerning univalent functions, folia societatis scientiarium lubliniensis, Mat. fiz. chem., 2, 57-58, (1962)
[12] Krzyż, J.; Lewandowski, Z., On the integral of univalent functions, Bull. acad. Pol. sci. ser. sci. math. astron. phys., 11, 447-448, (1963) · Zbl 0137.05202
[13] Lehto, O.; Virtanen, K.I., Quasiconformal mappings in the plane, () · Zbl 0267.30016
[14] Ozaki, S., On the theory of multivalent functions, Sci. rep. Tokyo bunrika daigaku A, 2, 167-188, (1935) · JFM 61.0353.02
[15] Pommerenke, Ch., Univalent functions, (1975), Vandenhoeck and Ruprecht Göttingen · Zbl 0283.30034
[16] Ponnusamy, S., On bazilevič functions, Ann. univ. mariae Curie — sklowdswska, vol XLII, 14, 115-127, (1988), (SECTIO A) · Zbl 0719.30009
[17] Ponnusamy, S., Differential subordination and starlike functions, Complex variables theory appl., 19, 185-194, (1992) · Zbl 0724.30020
[18] Ponnusamy, S., Univalence of Alexander transform under new mapping properties, Complex variables theory appl., 30, 1, 55-68, (1996) · Zbl 0865.30007
[19] Ponnusamy, S., Role of special functions in function theory, (), 127-137 · Zbl 0931.33003
[20] Ponnusamy, S., Close-to-convexity properties of Gaussian hypergeometric functions, J. comput. appl. math., 88, 327-337, (1997) · Zbl 0901.30007
[21] Ponnusamy, S., Inclusion theorems for convolution product of second order polylogarithms and functions with the derivative in a halfplane, Rocky mountain J. math., 28, 1, 39, (1998), to appear · Zbl 0915.30012
[22] Ponnusamy, S.; Rønning, F., Duality for Hadamard products applied to certain integral transforms, Complex variables theory appl., 32, 263-287, (1997) · Zbl 0878.30007
[23] Ponnusamy, S.; Rønning, F., Integral transforms of functions with the derivative in a half plane, (), 10 · Zbl 0943.30010
[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 of Gaussian hypergeometric functions, (), 34 · Zbl 0973.30017
[26] Ruscheweyh, St., Convolution in geometric function theory, (1982), Les Presses de l’Université de Montréal Montréal · Zbl 0575.30008
[27] Ruscheweyh, St.; Singh, V., On the order of starlikeness of hypergeometric functions, J. math. anal. appl., 113, 1-11, (1986) · Zbl 0598.30021
[28] Zmorovič, V.A., On some problems in the theory of univalent functions, Nauk. zapiski, kiev. derjavnyi univ., 83-94, (1952), (Russian)
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