Taiwanese Journal of Mathematics


Shigeyoshi Owa and A. A. Attiya

Full-text: Open access


For functions belonging to the generalization class $\mathcal{R}_{p}(k,\alpha ,\lambda )$ of analytic functions in the open unit disc $\mathbb{U}=\{z:\,\left| z\right| \lt 1\}$, which investigated in this paper, some applications of differential subordination are obtained which contain interesting property of Hurwitz-Lerch Zeta function.

Article information

Taiwanese J. Math., Volume 13, Number 2A (2009), 369-375.

First available in Project Euclid: 18 July 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.) 30C80: Maximum principle; Schwarz's lemma, Lindelöf principle, analogues and generalizations; subordination

analytic function univalent function convex function differential subordination Hurwitz-Lerch Zeta function


Owa, Shigeyoshi; Attiya, A. A. AN APPLICATION OF DIFFERENTIAL SUBORDINATIONS TO THE CLASS OF CERTAIN ANALYTIC FUNCTIONS. Taiwanese J. Math. 13 (2009), no. 2A, 369--375. doi:10.11650/twjm/1500405342. https://projecteuclid.org/euclid.twjm/1500405342

Export citation


  • M. P. Chen, On functions satisfying ${\rm Re}\left\{ f(z)/z\right\} >\alpha ,$ Tamkang J. Math., 5 (1974), 231-234.
  • M. P. Chen, On the regular functions satisfying ${\rm Re}\left\{ f(z)/z\right\} >\alpha ,$ Bull. Inst. Math. Acad. Sinica, 3 (1975), 65-70.
  • N. E. Cho, On certain classes of p-valent analytic functions, Int. J. Math. Math. Sci., 16 (1993), 319-328.
  • J. Choi and H. M. Srivastava, Certain families of series associated with the Hurwitz-Lerch Zeta functions, Appl. Math. Comput. 170 (2005), 399-409.
  • T. G. Èzrohi, Certain estimates in special class of univalent functions in the unit circle $\left| z\right| <1,$ Dopovidi Akad. Nauk. Ukrain RSR, (1965), 984-988.
  • C. Ferreira and J. L. López, Asymptotic expansions of the Hurwitz-Lerch zeta function, J. Math. Anal. Appl., 298 (2004), 210-224.
  • S. Fukui, S. Owa and M. Nunokawa, The radius of starlikeness of functions satisfying ${\rm Re}\left\{ f(z)/z^{p}\right\} >0.$ Bull. Fac. Ed. Wakayama Univ. Natur. Sci., 37 (1988), 11-13.
  • R. M. Goel, On functions satisfying ${\rm Re}\left\{ f(z)/z\right\} >\alpha ,$ Publ. Math. Debrecen, 18 (1971), 111-117.
  • R. M. Goel, The radius of convexity and starlikeness for certain classes of analytic functions with fixed second coefficient, Ann. Univ. Mariae Curie-Sklodowska Soci., 25 (1971), 33-39.
  • D. J. Hallenbeck and St. Ruscheweyh, Subordination by convex functions, Proc. Amer. Math. Soc., 52 (1975), 191-195.
  • S. D. Lin, H. M. Srivastava and P. Y. Wang, Some expansion formulas for a class of generalized Hurwitz-Lerch Zeta functions, Integral Transform. Spec. Funct., 17 (2006), 817-722.
  • Q. M. Luo, and H. M. Srivastava, Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials, J. Math. Anal. Appl., 308 (2005), 290-302.
  • T. H. MacGregor, Functions whose derivative has a positive real part, Trans. Amer. Math. Soc., 104 (1962), 532-537.
  • S. S. Miller and P. T. Mocanu, Differential Subordinations: Theory and Applications, Series in Pure and Applied Mathematics, No. 225. Marcel Dekker, Inc., New York, 2000.
  • H. Saitoh, On certain subclasses of analytic functions involving a linear operator. Transform Methods and Special Functions (in Varna '96), Bulg. Acad. Sci., (1998), 401-411.
  • H. Saitoh and M. Nunokawa, On functions satisfying $ Re\left\{
  • f(z)/z^p\right\} >0.$ {\em Bull. Soci. Roy. Sci. Li\`{e}ge}, {\bf 56} (1987), 462-464.\bibitem{SNOF} H. Saitoh, M. Nunokawa, S. Owa and S. Fukui, A note on certain class of analytic functions satisfying $ Re\left f(z)/z^p\right >0,$ Math. Today, 7 (1989), 33-36.
  • H. M. Srivastava and A. A. Attiya, An integral operator associated with the Hurwitz-Lerch Zeta function and differential subordination, Integral Transform. Spec. Funct., 18 (2007), 95-107.
  • H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, Boston, and London, 2001.
  • T. Umezawa, Multivalently close-to-convex functions, Proc. Amer. Math. Soc., 8 (1957), 869-874.
  • K. Yamaguchi, On functions satisfying ${\rm Re}\left\{ f(z)/z\right\} >\alpha ,$ Proc. Amer. Math. Soc., 17 (1966), 588-591.