Fekete-Szegő problem for subclasses of generalized uniformly starlike functions with respect to symmetric points

Nihat Yagmur; Halit Orhan

Mathematica Bohemica (2014)

  • Volume: 139, Issue: 3, page 485-509
  • ISSN: 0862-7959

Abstract

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The authors obtain the Fekete-Szegő inequality (according to parameters s and t in the region s 2 + s t + t 2 < 3 , s t and s + t 2 , or in the region s 2 + s t + t 2 > 3 , s t and s + t 2 ) for certain normalized analytic functions f ( z ) belonging to k -UST λ , μ n ( s , t , γ ) which satisfy the condition ( s - t ) z ( D λ , μ n f ( z ) ) ' D λ , μ n f ( s z ) - D λ , μ n f ( t z ) > k ( s - t ) z ( D λ , μ n f ( z ) ) ' D λ , μ n f ( s z ) - D λ , μ n f ( t z ) - 1 + γ , z 𝒰 . 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.

How to cite

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Yagmur, Nihat, and Orhan, Halit. "Fekete-Szegő problem for subclasses of generalized uniformly starlike functions with respect to symmetric points." Mathematica Bohemica 139.3 (2014): 485-509. <http://eudml.org/doc/261977>.

@article{Yagmur2014,
abstract = {The authors obtain the Fekete-Szegő inequality (according to parameters $s$ and $t$ in the region $s^\{2\}+st+t^\{2\}<3$, $s\ne t$ and $s+t\ne 2$, or in the region $s^\{2\}+st+t^\{2\}>3,$$s\ne t$ and $s+t\ne 2$) 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 \lbrace \frac\{(s-t)z ( D\_\{\lambda ,\mu \}^\{n\}f(z))^\{\prime \}\}\{D\_\{\lambda ,\mu \}^\{n\}f(sz)-D\_\{\lambda ,\mu \}^\{n\}f(tz)\}\bigg \rbrace >k \biggl \vert \frac\{(s-t)z ( D\_\{\lambda ,\mu \}^\{n\}f(z))^\{\prime \}\}\{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.},
author = {Yagmur, Nihat, Orhan, Halit},
journal = {Mathematica Bohemica},
keywords = {Fekete-Szegő problem; Sakaguchi function; uniformly starlike function; symmetric point; Fekete-Szegő problem; Sakaguchi function; uniformly starlike function; symmetric point},
language = {eng},
number = {3},
pages = {485-509},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Fekete-Szegő problem for subclasses of generalized uniformly starlike functions with respect to symmetric points},
url = {http://eudml.org/doc/261977},
volume = {139},
year = {2014},
}

TY - JOUR
AU - Yagmur, Nihat
AU - Orhan, Halit
TI - Fekete-Szegő problem for subclasses of generalized uniformly starlike functions with respect to symmetric points
JO - Mathematica Bohemica
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 139
IS - 3
SP - 485
EP - 509
AB - The authors obtain the Fekete-Szegő inequality (according to parameters $s$ and $t$ in the region $s^{2}+st+t^{2}<3$, $s\ne t$ and $s+t\ne 2$, or in the region $s^{2}+st+t^{2}>3,$$s\ne t$ and $s+t\ne 2$) 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 \lbrace \frac{(s-t)z ( D_{\lambda ,\mu }^{n}f(z))^{\prime }}{D_{\lambda ,\mu }^{n}f(sz)-D_{\lambda ,\mu }^{n}f(tz)}\bigg \rbrace >k \biggl \vert \frac{(s-t)z ( D_{\lambda ,\mu }^{n}f(z))^{\prime }}{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.
LA - eng
KW - Fekete-Szegő problem; Sakaguchi function; uniformly starlike function; symmetric point; Fekete-Szegő problem; Sakaguchi function; uniformly starlike function; symmetric point
UR - http://eudml.org/doc/261977
ER -

References

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  1. Al-Oboudi, F. M., 10.1155/S0161171204108090, Int. J. Math. Math. Sci. 2004 1429-1436 (2004). (2004) Zbl1072.30009MR2085011DOI10.1155/S0161171204108090
  2. Al-Oboudi, F. M., Al-Amoudi, K. A., 10.1016/j.jmaa.2007.05.087, J. Math. Anal. Appl. 339 655-667 (2008). (2008) Zbl1132.30010MR2370683DOI10.1016/j.jmaa.2007.05.087
  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). (1997) Zbl0898.30010MR1457247
  4. Cho, N. E., Kwon, O. S., Owa, S., Certain subclasses of Sakaguchi functions, Southeast Asian Bull. Math. 17 121-126 (1993). (1993) Zbl0788.30007MR1259988
  5. Deniz, E., Çağlar, M., Orhan, H., 10.2996/kmj/1352985448, Kodai Math. J. 35 439-462 (2012). (2012) Zbl1276.30022MR2997474DOI10.2996/kmj/1352985448
  6. Deniz, E., Orhan, H., 10.5666/KMJ.2010.50.1.037, Kyungpook Math. J. 50 37-47 (2010). (2010) Zbl1200.30010MR2609085DOI10.5666/KMJ.2010.50.1.037
  7. Gangadharan, A., Shanmugam, T. N., Srivastava, H. M., 10.1016/S0898-1221(02)00275-4, Comput. Math. Appl. 44 1515-1526 (2002). (2002) Zbl1036.33003MR1944665DOI10.1016/S0898-1221(02)00275-4
  8. Goodman, A. W., On uniformly convex functions, Ann. Pol. Math. 56 87-92 (1991). (1991) Zbl0744.30010MR1145573
  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). (2012) Zbl1251.30015MR2952858
  10. Kanas, S., 10.1016/j.amc.2012.01.070, Appl. Math. Comput. 218 8453-8461 (2012). (2012) Zbl1251.30018MR2921337DOI10.1016/j.amc.2012.01.070
  11. Kanas, S., Darwish, H. E., 10.1016/j.aml.2010.03.008, Appl. Math. Lett. 23 777-782 (2010). (2010) Zbl1189.30021MR2639878DOI10.1016/j.aml.2010.03.008
  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). (1990) Zbl0741.30012MR1114742
  13. Kanas, S., Srivastava, H. M., 10.1080/10652460008819249, Integral Transforms Spec. Funct. 9 121-132 (2000). (2000) Zbl0959.30007MR1784495DOI10.1080/10652460008819249
  14. Kanas, S., Sugawa, T., On conformal representations of the interior of an ellipse, Ann. Acad. Sci. Fenn., Math. 31 329-348 (2006). (2006) Zbl1098.30011MR2248819
  15. Kanas, S., Wiśniowska, A., Conic domains and starlike functions, Rev. Roum. Math. Pures Appl. 45 647-657 (2000). (2000) Zbl0990.30010MR1836295
  16. Kanas, S., Wisniowska, A., 10.1016/S0377-0427(99)00018-7, J. Comput. Appl. Math. 105 327-336 (1999). (1999) Zbl0944.30008MR1690599DOI10.1016/S0377-0427(99)00018-7
  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). (1994) Zbl0823.30007MR1343506
  18. Mishra, A. K., Gochhayat, P., 10.2996/kmj/1278076345, Kodai Math. J. 33 310-328 (2010). (2010) Zbl1196.30013MR2681543DOI10.2996/kmj/1278076345
  19. Mishra, A. K., Gochhayat, P., 10.1016/j.jmaa.2008.06.009, J. Math. Anal. Appl. 347 563-572 (2008). (2008) MR2440350DOI10.1016/j.jmaa.2008.06.009
  20. Orhan, H., Deniz, E., Raducanu, D., 10.1016/j.camwa.2009.07.049, Comput. Math. Appl. 59 283-295 (2010). (2010) Zbl1189.30049MR2575514DOI10.1016/j.camwa.2009.07.049
  21. Orhan, H., Gunes, E., Fekete-Szegő inequality for certain subclass of analytic functions, Gen. Math. 14 41-54 (2006). (2006) Zbl1164.30345MR2233678
  22. Orhan, H., Răducanu, D., 10.1016/j.mcm.2009.04.014, Math. Comput. Modelling 50 430-438 (2009). (2009) Zbl1185.30014MR2542789DOI10.1016/j.mcm.2009.04.014
  23. Orhan, H., Yagmur, N., Çağlar, M., 10.5644/SJM.08.2.05, Sarajevo J. Math. 8(21) 235-244 (2012). (2012) MR3057883DOI10.5644/SJM.08.2.05
  24. Owa, S., Sekine, T., Yamakawa, R., Notes on Sakaguchi functions, Aust. J. Math. Anal. Appl. (electronic only) 3 Article 12, 7 pages (2006). (2006) Zbl1090.30024MR2223016
  25. Owa, S., Sekine, T., Yamakawa, R., 10.1016/j.amc.2006.08.133, Appl. Math. Comput. 187 356-361 (2007). (2007) Zbl1113.30018MR2323589DOI10.1016/j.amc.2006.08.133
  26. Owa, S., Srivastava, H. M., 10.4153/CJM-1987-054-3, Can. J. Math. 39 1057-1077 (1987). (1987) Zbl0611.33007MR0918587DOI10.4153/CJM-1987-054-3
  27. Răducanu, D., Orhan, H., Subclasses of analytic functions defined by a generalized differential operator, Int. J. Math. Anal., Ruse 4 1-15 (2010). (2010) Zbl1195.30031MR2657755
  28. Rønning, F., 10.1090/S0002-9939-1993-1128729-7, Proc. Am. Math. Soc. 118 189-196 (1993). (1993) MR1128729DOI10.1090/S0002-9939-1993-1128729-7
  29. Sakaguchi, K., 10.2969/jmsj/01110072, J. Math. Soc. Japan 11 72-75 (1959). (1959) Zbl0085.29602MR0107005DOI10.2969/jmsj/01110072
  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). (1983) Zbl0531.30009MR0738107
  31. Srivastava, H. M., Mishra, A. K., 10.1016/S0898-1221(99)00333-8, Comput. Math. Appl. 39 57-69 (2000). (2000) Zbl0948.30018MR1740907DOI10.1016/S0898-1221(99)00333-8
  32. Srivastava, H. M., Mishra, A. K., Das, M. K., 10.1080/17476930108815351, Complex Variables, Theory Appl. 44 145-163 (2001). (2001) Zbl1021.30014MR1908584DOI10.1080/17476930108815351

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