On a certain class of harmonic functions and the generalized Bernardi-Libera-Livingston integral operator

Grigore Stefan Salagean, Pall-Szabo Agnes Orsolya


In this paper we examine the closure properties of the class $\mathcal{V}_\mathcal{H}(F;\gamma)$ under the generalized Bernardi-Libera-Livingston integral \\operator $\mathcal{L}_{c}(f),$ ($c>-1$) which is defined by
$ \mathcal{L}_{c}(f)=\mathcal{L}_{c}(h)+\overline{\mathcal{L}_{c}(g)}$ where
\mathcal{L}_{c}(h)(z)=\frac{c+1}{z^{c}}\int\limits_{0}^{z}(t^{c-1}h(t) dt \;\;\;%
\mathrm{and} \;\;\; \mathcal{L}_{c}(g)(z)=\frac{c+1}{z^{c}}\int\limits_{0}^{z}(t^{c-1}g(t) dt.
The obtained results are sharp and they improve known results.


harmonic univalent functions; extreme points

Full Text:



B.C. Carlson, S.B. Shaffer, Starlike and prestarlike hypergeometric functions, SIAM, J. Math. Anal., 15 (2002), 737-745.

J. Clunie, T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser.A.I. Math., 9 (1984) 3-25.

J.M. Jahangiri, G. Murugusundaramoorthy, K. Vijaya, Salagean-type harmonic univalent functions, Southwest. J. Pure and Apll., Math., 2 (2002), 77-82.

J.M. Jahangiri, G. Murugusundaramoorthy, K.Vijaya, Starlikeness of

Rucheweyh type harmonic univalent functions, J. Indian Acad. Math., 26 (2004), 191-200.

J.M. Jahangiri, H. Silverman, Harmonic univalent functions with varying arguments, Internat. J. Appl. Math., 8 (2002), 267-275.

H.A. Al-Kharsani, R.A. Al-Khai, Univalent harmonic functions, J.Ineqal. Pure and Appl.Maths., 8 Issue 2, Articol 59 , 8 pp.

G. Murugusundaramoorthy, A class of Ruscheweyh-type harmonic univalent functions with varying arguments, Southwest J. Pure Appl. Math., 2 (2003), 90-95.

G. Murugusundaramoorthy, G.S.Salagean , On a certain class of harmonic functions associated with a convolution structure, Mathematica, Tome 54 (77),no special 2012, pp. 131-142

G. Murugusundaramoorthy, K. Vijaya, On certain subclasses of harmonic functions associated with Wright hypergeometric functions, Advanced Studies Contemporary Mathematics, 1 (2009).

S. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc., 49 (1975), 109 -115.

G. S. Salagean, Subclasses of univalent functions, Complex analysis-fifth Romanian-Finnish seminar, Part 1 (Bucharest, 1981), Lecture Notes in Math., 1013, Springer, Berlin, (1983), 362-372.

H.M. Srivastava, S. Owa, Some characterization and distortion theorems involving fractional calculus, generalized hypergeometric functions, Hadamard products, linear operators and certain subclasses of analytic functions, Nagoya Math. J., 106 (1987), 1-28.

E.M.Wright, The asymptotic expansion of the generalized hypergeometric function, Proc. London. Math. Soc., 46 (1946), 389-408.

DOI: http://dx.doi.org/10.24193/subbmath.2020.3.05


  • There are currently no refbacks.