Abstract
Using the idea of convolution between analytic functions, we define a class \(\mathcal {UM}(g,\gamma ,b,k)\) of analytic functions comprising of starlike and convex functions. These functions map the open unit disc on to the conic domains. We derive some sufficient conditions and then use them to define the class \(\mathcal {UM}^{*}(g,\gamma ,b,k)\). Making use of an increasing factor sequence, we discuss a subordination result. We may relate our findings with the previously known results.
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Ahuja, O., Murugusundaramoorthy, G., Magesh, N.: Integral means for uniformly convex and starlike functions associated with generalized hypergeometric functions. J. Inequal. Pure Appl. Math. 8(4), 1–9 (2007)
Al-Refai, O., Darus, M.: An extension to the Owa–Srivastava fractional operator with applications to parabolic starlike and uniformly convex functions. Int. J. Differ. Equ. Art. ID 597292, 18 Pages (2009)
Aouf, M.K., El-Ashwah, R.M., El-Deeb, S.M.: Subordination results for certain subclasses of uniformly starlike and convex functions defined by convolution. Eur. J. Pure Appl. Math. 3(5), 903–917 (2010)
Aouf, M.K., Mostafa, A.O.: Some properties of a subclass of uniformly convex functions with negative coefficients. Demonstr. Math. 2, 353–370 (2008)
Attiya, A.A.: On some application of a subordination theorems. J. Math. Anal. Appl. 311, 489–494 (2005)
Bapna, I.B., Nidhi, J.: Parabolic starlike and uniformly convex functions. Int. J. Math. Anal. 4, 1913–1921 (2010)
Bernardi, S.D.: Convex and starlike univalent functions. Trans. Am. Math. Soc. 135, 429–446 (1969)
Bharati, R., Parvatham, R., Swaminathan, A.: On subclasses of uniformly convex functions and corresponding class of starlike functions. Tamakang J. Math. 28, 17–32 (1997)
Cho, N.E., Kwon, O.S., Srivastava, H.M.: Inclusion relationships and argument properties for certain subclasses of multivalent functions associated with a family of linear operators. J. Math. Anal. Appl. 292, 470–483 (2004)
Choi, J.H., Saigo, M., Srivastava, H.M.: Some inclusion properties of a certain family of integral operators. J. Math. Anal. Appl. 276, 432–445 (2002)
Dixit, K.K., Verma, V.: Uniformly starlike and uniformly convexity properties for hypergeometric functions. Bull. Cal. Math. Soc. 93(6), 477–482 (2001)
Frasin, B.A.: Subordination results for a class of analytic functions defined by a linear operator. J. Inequal. Pure Appl. Math. 7(4), 1–7 (2006)
Goodman, A.W.: On uniformly convex functions. Ann. Polon. Math. 56, 87–92 (1991)
Goodman, A.W.: On uniformly starlike functions. J. Math. Anal. Appl. 155, 364–370 (1991)
Gupta, K., Jain, C.R., Sharma, A.: A study of unified finite integral transforms with applications. J. Rajasthan Acad. Phys. Sci. 2(4), 269–282 (2003)
Kanas, S., Srivastava, H.M.: Linear operators associated with k-uniformly convex functions. Integr. Transforms Spec. Funct. 9, 121–132 (2000)
Kanas, S., Wiśniowska, A.: Conic regions and k-uniform convexity. J. Appl. Math. 105, 327–336 (1999)
Kanas, S., Wiśniowska, A.: Conic domains and starlike functions. Rev. Roumaine Math. Pures Appl. 45, 647–657 (2000)
Libera, R.J.: Some classes of regular univalent functions. Proc. Am. Math. Soc. 16, 755–758 (1965)
Ma, W.C., Minda, D.: Uniformly convex functions. Ann. Polon. Math. 57, 165–175 (1992)
Miller, S.S., Mocanu, P.T.: Differenatial Subordinations: Theory and Applications. Series on Monographs and Textbooks in Pure and Appl Math, vol. 255. Marcel Dekker, Inc, New York (2000)
Murugusundaramoorthy, G., Magesh, N.: A new subclass of uniformly convex functions and corresponding subclass of starlike functions with fixed second coefficient. J. Inequal. Pure Appl. Math. 5(4), 1–10 (2004)
Murugusundaramoorthy, G., Magesh, N.: Linear operators associated with a subclass of uniformly convex functions. Int. J. Pure Appl. Math. Sci. 4, 113–125 (2006)
Murugusundaramoorthy, G., Rosy, T., Muthunagai, G.: Carlson–Shaffer operator and their applications to certain subclass of uniformly convex function. Gen. Math. 15, 131–143 (2007)
Noor, K.I.: On new classes of integral operators. J. Nat. Geom. 16, 71–80 (1999)
Raina, R.K., Deepak, B.: Some properties of a new class of analytic functions defined in terms of a hadamard product. J. Inequal. Pure Appl. Math. 9, 1–9 (2008)
Ronning, F.: On starlike functions associated with parabolic regions. Ann. Univ. Mariae Curie-Sklodowska Sect. A. 45, 117–122 (1991)
Ronning, F.: Uniformly convex functions and a corresponding class of starlike functions. Proc. Am. Math. Soc. 1(18), 189–196 (1993)
Rosy, T., Murugusundaramoorthy, G.: Fractional calculus and their applications to certain subclass of uniformly convex functions. Far East J. Math. Sci. 15, 231–242 (2004)
Ruscheweyh, St: New criteria for univalent functions. Proc. Am. Math. Soc. 49, 109–115 (1975)
Srivastava, H.M., Attiya, A.A.: Some subordination results associated with certain subclass of analytic functions. J. Inequal. Pure Appl. Math. 5(4), 1–6 (2004)
Srivastava, H.M., Mishra, A.K.: Applications of fractional calculus to parabolic starlike and uniformly convex functions. Comput. Math. Appl. 39, 57–69 (2000)
Subramanian, K.G., Murugusundaramoorthy, G., Balasubrahma, P., Silverman, H.: Subclasses of uniformly convex and uniformly starlike functions. Math. Japon. 42, 517–522 (1995)
Wilf, H.S.: Subordinating factor sequence for convex maps of the unit circle. Proc. Am. Math. Soc. 129, 689–693 (1961)
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Bukhari, S.Z.H., Sokol, J. & Zafar, S. Unified Approach to Starlike and Convex Functions Involving Convolution Between Analytic Functions. Results Math 73, 30 (2018). https://doi.org/10.1007/s00025-018-0782-0
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DOI: https://doi.org/10.1007/s00025-018-0782-0