Integral Operator of Analytic Functions with Positive Real Part

  • Frasin, Basem Aref
  • Received : 2010.08.03
  • Accepted : 2010.09.27
  • Published : 2011.03.31


In this paper, we introduce the integral operator $I_{\beta}$($p_1$, ${\ldots}$, $p_n$; ${\alpha}_1$, ${\ldots}$, ${\alpha}_n$)(z) analytic functions with positive real part. The radius of convexity of this integral operator when ${\beta}$ = 1 is determined. In particular, we get the radius of starlikeness and convexity of the analytic functions with Re {f(z)/z} > 0 and Re {f'(z)} > 0. Furthermore, we derive sufficient condition for the integral operator $I_{\beta}$($p_1$, ${\ldots}$, $p_n$; ${\alpha}_1$, ${\ldots}$, ${\alpha}_n$)(z) to be analytic and univalent in the open unit disc, which leads to univalency of the operators $\int\limits_0^z(f(t)/t)^{\alpha}$dt and $\int\limits_0^z(f'(t))^{\alpha}dt$.


Analytic and univalent functions;Starlike and convex functions;Functions of positive real part;Integral operator


  1. W. Alexander, Functions which map the interior of the unit circle upon simple regions, Anna. Math., 17(1915), 12-22.
  2. F. M. Al-Oboudi, On univalent functions defined by a generalized Salagean operator, Int. J. Math. Math. Sci., 27(2004), 1429-1436.
  3. D. Breaz and N. Breaz, Two integral operator, Studia Universitatis Babes-Bolyai, Mathematica, Cluj-Napoca, 3(2002), 13-21.
  4. D. Breaz, H. Guney and G. Salagean, A new integral operator, Tamsui Oxford Journal of Mathematical Sciences, 25(4)(2009), 407-414.
  5. D. Breaz, S. Owa and N. Breaz, A new integral univalent operator, Acta Univ. Apul., 16(2008), 11-16.
  6. S. Bulut, Some properties for an integral operator defined by Al-Oboudi differential operator, JIPAM, 9(2008), Issue 4, Atr. 115, 5 pp.
  7. S. Bulut, Univalence preserving integral operators defined by generalized Al-Oboudi differential operators, An. St. Univ. Ovidius Constata, 17(2009), 37-50.
  8. B. C. Carlson, D. B. Shaffer, Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal., 15(4)(1984), 737-745.
  9. B. A. Frasin, General integral operator de ned by Hadamard product, Math Vesnik, 62, 2(2010), 127-136.
  10. I. J. Kim and E. P. Merkes, On an integral of powers of a spirallike function, Kyungpook Math. J., 12(2)(1972), 249-253.
  11. G. I. Oros, G. Oros and D. Breaz, Sufficient conditions for univalence of an integral operator, J. Ineq. Appl., Vol. 2008, Article ID 127645, 7 pages.
  12. T. MacGregor, The radius of univalence of certain analytic functions, Proc. Amer. Math. Soc., 14(1963), 514-520.
  13. S. S. Miller, P. T. Mocanu, and M. O. Reade, Starlike integral operators, Pacific J. Math., 79(1978), 157-168.
  14. N. Pascu, An improvement of Backer's univalence criterion, Proceedings of the Commemorative Session Simion Stoilow, Brasov, (1987), 43-48.
  15. N. Pascu and V. Pescar, On the integral operators of Kim-Merkis and Pfaltzgraff, Mathematica, 32(55)(1990), 185-192.
  16. J. A. Pfaltzgraff, Univalence of the integral of $f^{'}(z)^{\lambda}$, Bull. London Math. Soc., 7(3)(1975), 254-256.
  17. St. Ruscheweyh, New criteria for univalent functions, Proc. Amer.Math. Soc., 49(1975), 109-115.
  18. G. Salagean, Subclasses of univalent functions, Lecture Notes in Math., (Springer-Verlag), 1013(1983), 362-372.
  19. C. Selvaraj and K. R. Karthikeyan, Sufficient conditions for univalence of a general integral operator, Bull. Korean Math. Soc., 46(2)(2009), 367-372.

Cited by

  1. General Integral Operator of Analytic Functions Involving Functions with Positive Real Part vol.2013, 2013,
  2. On General Integral Operator of Analytic Functions vol.2013, 2013,