International Journal of Nonlinear Analysis and Applications
A unified class of analytic functions associated with Erdeyi-Kober type integral operator
Document Type : Research Paper
Authors
School of Advanced Sciences, Vellore Institute of Technology, Vellore-632014, TN, India
Abstract
Making use of the convolution product, we introduce a unified class of analytic functions with negative coefficients. Also, we obtain the coefficient bounds, extreme points and radius of starlikeness for functions belonging to the generalized class $ \mathcal{TP}_{\vartheta,\tau}^{a,c}(\alpha ,\beta).$ Furthermore, partial sums $f_k(z)$ of functions $f(z)$ in the class $\mathcal{P}_{\vartheta,\tau}^{a,c}(\alpha ,\beta)$ are considered and sharp lower bounds for the ratios of real part of $f(z)$ to $f_k(z)$ and $f'(z)$ to $f'_k(z)$ are determined. Relevant connections of the results with various known results are also considered.
Keywords
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[11] I.B. Jung, Y.C. Kim and H.M. Srivastava, The Hardy space of analytic functions associated with certain oneparameter families of integral operators, J. Math. Anal. Appl. 176, 138–147, 1993
[12] V. Kiryakova, Generalized fractional calculus and applications, Pitman Research Notes in Mathematics Series, 301, John Willey & Sons, Inc. New York, 1994.
[13] R.J. Libera, Some classes of regular univalent functions, Proc. Amer. Math. Soc. 16 (1965), 755–758.
[14] A.E. Livingston, On the radius of univalence of certain analytic functions, Proc. Amer. Math. Soc. 17 (1966), 352–357.
[15] K.I. Noor, On new classes of integral operators, J. Natural Geometry 16 (1999), 71–80.
[16] K.I. Noor and M.A. Noor, On integral operators, J. Math. Anal. Appl. 238 (1999), 341–352.
[17] J.E. Littlewood, On inequalities in theory of functions, Proc. London Math. Soc. 23 (1925), 481–519.
[18] F. Rønning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc. 118 (1993), 189–196.
[19] F. Rønning, Integral representations for bounded starlike functions, Annal. Polon. Math. 60 (1995), 289–297.
[20] T. Rosy, K.G. Subramanian and G. Murugusundaramoorthy, Neighbourhoods and partial sums of starlike functions based on Ruscheweyh derivatives, J. Ineq. Pure Appl. Math. 4 (2003), no. 4.
[21] T. Rosy and G.Murugusundaramoorthy, Fractional calculus and their applications to certain subclass of uniformly convex functions, Far East. J. Math. Sci. 15 (2004), no. 2, 231–242.
[22] S. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc. 49 (1975), 109–115.
[23] E.M. Silvia, Partial sums of convex functions of order α, Houston J. Math. 11 (1985), no. 3, 397–404.
[24] H. Silverman, Univalent functions with negative coefficients, Proc. Amer. Math. Soc. 51 (1975), 109–116.
[25] H. Silverman, A survey with open problems on univalent functions whose coefficients are negative, Rocky Mt. J. Math., 21 (1991), 1099–1125.
[26] H. Silverman, Integral means for univalent functions with negative coefficients, Houston J. Math. 23 (1997), 169–174.
[27] H. Silverman, Partial sums of starlike and convex functions, J. Math.Anal. Appl. 209 (1997), 221–227.
[28] K.G. Subramanian, G. Murugusundaramoorthy, P. Balasubrahmanyam and H. Silverman, Subclasses of uniformly convex and uniformly starlike functions. Math. Japonica 42 (1995), no. 3, 517–522.
[29] K.G. Subramanian, T.V. Sudharsan, P. Balasubrahmanyam and H.Silverman, Classes of uniformly starlike functions. Publ. Math. Debrecen 53 (1998), no. 3-4, 309–315 .
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Volume 14, Issue 2
February 2023Pages 385-397
- Receive Date: 19 August 2021
- Revise Date: 09 November 2021
- Accept Date: 12 November 2021