Integral Operator of Analytic Functions with Positive Real Part

• Journal title : Kyungpook mathematical journal
• Volume 51, Issue 1,  2011, pp.77-85
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2011.51.1.077
Title & Authors
Integral Operator of Analytic Functions with Positive Real Part
Frasin, Basem Aref;

Abstract
In this paper, we introduce the integral operator $\small{I_{\beta}}$($\small{p_1}$, $\small{{\ldots}}$, $\small{p_n}$; $\small{{\alpha}_1}$, $\small{{\ldots}}$, $\small{{\alpha}_n}$)(z) analytic functions with positive real part. The radius of convexity of this integral operator when $\small{{\beta}}$ = 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 $\small{I_{\beta}}$($\small{p_1}$, $\small{{\ldots}}$, $\small{p_n}$; $\small{{\alpha}_1}$, $\small{{\ldots}}$, $\small{{\alpha}_n}$)(z) to be analytic and univalent in the open unit disc, which leads to univalency of the operators $\small{\int\limits_0^z(f(t)/t)^{\alpha}}$dt and $\small{\int\limits_0^z(f$.
Keywords
Analytic and univalent functions;Starlike and convex functions;Functions of positive real part;Integral operator;
Language
English
Cited by
1.
On General Integral Operator of Analytic Functions, Abstract and Applied Analysis, 2013, 2013, 1
2.
General Integral Operator of Analytic Functions Involving Functions with Positive Real Part, Journal of Mathematics, 2013, 2013, 1
References
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.