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Article

Yagmur, Nihat ; Orhan, Halit
Fekete-Szegő problem for subclasses of generalized uniformly starlike functions with respect to symmetric points. (English). Mathematica Bohemica, vol. 139 (2014), issue 3, pp. 485-509
MSC: 30C45, 30C50 | MR 3269370 | Zbl 06391467 | DOI: 10.21136/MB.2014.143938
Keywords:
Fekete-Szeg\H {o} problem; Sakaguchi function; uniformly starlike function; symmetric point
Summary:
The authors obtain the Fekete-Szegő inequality (according to parameters $s$ and $t$ in the region $s^{2}+st+t^{2}<3$, $s\neq t$ and $s+t\neq 2$, or in the region $s^{2}+st+t^{2}>3,$ $s\neq t$ and $s+t\neq 2$) for certain normalized analytic functions $f(z)$ belonging to $k\text {\rm -UST}_{\lambda ,\mu }^{n}(s,t,\gamma )$ which satisfy the condition \begin {equation*} \Re \bigg \{ \frac {(s-t)z ( D_{\lambda ,\mu }^{n}f(z))'} {D_{\lambda ,\mu }^{n}f(sz)-D_{\lambda ,\mu }^{n}f(tz)}\bigg \} >k \biggl \vert \frac {(s-t)z ( D_{\lambda ,\mu }^{n}f(z))'}{D_{\lambda ,\mu }^{n}f(sz)-D_{\lambda ,\mu }^{n}f(tz)}{-1} \biggr \vert +\gamma , \quad z\in \mathcal {U} . \end {equation*} Also certain applications of the main result a class of functions defined by the Hadamard product (or convolution) are given. As a special case of this result, the Fekete-Szegő inequality for a class of functions defined through fractional derivatives is obtained.
References:
[1] Al-Oboudi, F. M.: On univalent functions defined by a generalized Sălăgean operator. Int. J. Math. Math. Sci. 2004 1429-1436 (2004). DOI 10.1155/S0161171204108090 | MR 2085011 | Zbl 1072.30009
[2] Al-Oboudi, F. M., Al-Amoudi, K. A.: On classes of analytic functions related to conic domains. J. Math. Anal. Appl. 339 655-667 (2008). DOI 10.1016/j.jmaa.2007.05.087 | MR 2370683 | Zbl 1132.30010
[3] Bharati, R., Parvatham, R., Swaminathan, A.: On subclasses of uniformly convex functions and corresponding class of starlike functions. Tamkang J. Math. 28 17-32 (1997). MR 1457247 | Zbl 0898.30010
[4] Cho, N. E., Kwon, O. S., Owa, S.: Certain subclasses of Sakaguchi functions. Southeast Asian Bull. Math. 17 121-126 (1993). MR 1259988 | Zbl 0788.30007
[5] Deniz, E., Çağlar, M., Orhan, H.: The Fekete-Szegő problem for a class of analytic functions defined by Dziok-Srivastava operator. Kodai Math. J. 35 439-462 (2012). DOI 10.2996/kmj/1352985448 | MR 2997474 | Zbl 1276.30022
[6] Deniz, E., Orhan, H.: The Fekete-Szegő problem for a generalized subclass of analytic functions. Kyungpook Math. J. 50 37-47 (2010). DOI 10.5666/KMJ.2010.50.1.037 | MR 2609085 | Zbl 1200.30010
[7] Gangadharan, A., Shanmugam, T. N., Srivastava, H. M.: Generalized hypergeometric functions associated with $k$-uniformly convex functions. Comput. Math. Appl. 44 1515-1526 (2002). DOI 10.1016/S0898-1221(02)00275-4 | MR 1944665 | Zbl 1036.33003
[8] Goodman, A. W.: On uniformly convex functions. Ann. Pol. Math. 56 87-92 (1991). DOI 10.4064/ap-56-1-87-92 | MR 1145573 | Zbl 0744.30010
[9] Goyal, S. P., Vijaywargiya, P., Darus, M.: Fekete-Szegő problem for subclasses of uniformly starlike functions with respect to symmetric points. Far East J. Math. Sci. (FJMS) 60 169-192 (2012). MR 2952858 | Zbl 1251.30015
[10] Kanas, S.: An unified approach to the Fekete-Szegő problem. Appl. Math. Comput. 218 8453-8461 (2012). DOI 10.1016/j.amc.2012.01.070 | MR 2921337 | Zbl 1251.30018
[11] Kanas, S., Darwish, H. E.: Fekete-Szegő problem for starlike and convex functions of complex order. Appl. Math. Lett. 23 777-782 (2010). DOI 10.1016/j.aml.2010.03.008 | MR 2639878 | Zbl 1189.30021
[12] Kanas, S., Lecko, A.: On the Fekete-Szegő problem and the domain of convexity for a certain class of univalent functions. Zesz. Nauk. Politech. Rzeszowskiej, Mat. Fiz. 10, Mat. 9 73 49-57 (1990). MR 1114742 | Zbl 0741.30012
[13] Kanas, S., Srivastava, H. M.: Linear operators associated with $k$-uniformly convex functions. Integral Transforms Spec. Funct. 9 121-132 (2000). DOI 10.1080/10652460008819249 | MR 1784495 | Zbl 0959.30007
[14] Kanas, S., Sugawa, T.: On conformal representations of the interior of an ellipse. Ann. Acad. Sci. Fenn., Math. 31 329-348 (2006). MR 2248819 | Zbl 1098.30011
[15] Kanas, S., Wiśniowska, A.: Conic domains and starlike functions. Rev. Roum. Math. Pures Appl. 45 647-657 (2000). MR 1836295 | Zbl 0990.30010
[16] Kanas, S., Wisniowska, A.: Conic regions and $k$-uniform convexity. J. Comput. Appl. Math. 105 327-336 (1999). DOI 10.1016/S0377-0427(99)00018-7 | MR 1690599 | Zbl 0944.30008
[17] Ma, W., Minda, D.: A unified treatment of some special classes of univalent functions. {Proceedings of the Conference on Complex Analysis, 1992, the Nankai Institute of Mathematics, Tianjin, China} Z. Li et al. Conf. Proc. Lecture Notes Anal. I International Press, Cambridge 157-169 (1994). MR 1343506 | Zbl 0823.30007
[18] Mishra, A. K., Gochhayat, P.: Fekete-Szegő problem for a class defined by an integral operator. Kodai Math. J. 33 310-328 (2010). DOI 10.2996/kmj/1278076345 | MR 2681543 | Zbl 1196.30013
[19] Mishra, A. K., Gochhayat, P.: The Fekete-Szegő problem for $k$-uniformly convex functions and for a class defined by the Owa-Srivastava operator. J. Math. Anal. Appl. 347 563-572 (2008). DOI 10.1016/j.jmaa.2008.06.009 | MR 2440350
[20] Orhan, H., Deniz, E., Raducanu, D.: The Fekete-Szegő problem for subclasses of analytic functions defined by a differential operator related to conic domains. Comput. Math. Appl. 59 283-295 (2010). DOI 10.1016/j.camwa.2009.07.049 | MR 2575514 | Zbl 1189.30049
[21] Orhan, H., Gunes, E.: Fekete-Szegő inequality for certain subclass of analytic functions. Gen. Math. 14 41-54 (2006). MR 2233678 | Zbl 1164.30345
[22] Orhan, H., Răducanu, D.: Fekete-Szegő problem for strongly starlike functions associated with generalized hypergeometric functions. Math. Comput. Modelling 50 430-438 (2009). DOI 10.1016/j.mcm.2009.04.014 | MR 2542789 | Zbl 1185.30014
[23] Orhan, H., Yagmur, N., Çağlar, M.: Coefficient estimates for Sakaguchi type functions. Sarajevo J. Math. 8(21) 235-244 (2012). DOI 10.5644/SJM.08.2.05 | MR 3057883
[24] Owa, S., Sekine, T., Yamakawa, R.: Notes on Sakaguchi functions. Aust. J. Math. Anal. Appl. (electronic only) 3 Article 12, 7 pages (2006). MR 2223016 | Zbl 1090.30024
[25] Owa, S., Sekine, T., Yamakawa, R.: On Sakaguchi type functions. Appl. Math. Comput. 187 356-361 (2007). DOI 10.1016/j.amc.2006.08.133 | MR 2323589 | Zbl 1113.30018
[26] Owa, S., Srivastava, H. M.: Univalent and starlike generalized hypergeometric functions. Can. J. Math. 39 1057-1077 (1987). DOI 10.4153/CJM-1987-054-3 | MR 0918587 | Zbl 0611.33007
[27] Răducanu, D., Orhan, H.: Subclasses of analytic functions defined by a generalized differential operator. Int. J. Math. Anal., Ruse 4 1-15 (2010). MR 2657755 | Zbl 1195.30031
[28] Rønning, F.: Uniformly convex functions and a corresponding class of starlike functions. Proc. Am. Math. Soc. 118 189-196 (1993). DOI 10.1090/S0002-9939-1993-1128729-7 | MR 1128729
[29] Sakaguchi, K.: On a certain univalent mapping. J. Math. Soc. Japan 11 72-75 (1959). DOI 10.2969/jmsj/01110072 | MR 0107005 | Zbl 0085.29602
[30] Sălăgean, G. S.: Subclasses of univalent functions. Complex Analysis---fifth Romanian-Finnish Seminar, Part 1, Bucharest, 1981 C. Andreian Cazacu et al. Lecture Notes in Math. 1013 Springer, Berlin 362-372 (1983). MR 0738107 | Zbl 0531.30009
[31] Srivastava, H. M., Mishra, A. K.: Applications of fractional calculus to parabolic starlike and uniformly convex functions. Comput. Math. Appl. 39 57-69 (2000). DOI 10.1016/S0898-1221(99)00333-8 | MR 1740907 | Zbl 0948.30018
[32] Srivastava, H. M., Mishra, A. K., Das, M. K.: The Fekete-Szegő problem for a subclass of close-to-convex functions. Complex Variables, Theory Appl. 44 145-163 (2001). DOI 10.1080/17476930108815351 | MR 1908584 | Zbl 1021.30014
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