Some applications of certain integral operators involving functions
Keywords:Convolution, hypergeometric functions, integral operator, differential subordination.
AbstractIntegral transforms map equations from their original domains into others where manipulations and solutions may be much easier than in original domains. To get back in the original environment, we use the idea of inverse of the integral transform. A function analytic and locally univalent in a given simply connected domain is said to be of bounded boundary rotation if its range has bounded boundary rotation which is defined as the total variation of the direction angle of the tangent to the boundary curve under a complete circuit. \qquad The main objective of the present article is to study some applications of certain integral operators to functions of bounded radius rotation involving Janowski functions. We discuss some inclusion results under certain assumption on parameters involve in operators as well as in related subclasses of analytic functions. Most of these results are best possible. We also relate our findings with the existing literature of the subjects.
F. S. M. Al Sarari and S. Latha, On symmetrical functions with bounded boundary rotation, J. Math. Comput. Sci., 4(3)(2014), 494—502.
S. Z. H. Bukhari and K. I. Noor, Some subclasses of analytic functions with bounded Mocanu variation, Acta Math. Hungarica, 126(3)(2010), 199--208.
A. Cetinkaya, Y. Kahramaner and Y.
Polatouglu ,Harmonic mappings related to the bounded boundary rotation, Int. J.Math. Anal., 8(57)(2014), 2837--2843.
S. D. Bernardi, The radius of univalence of certain analytic functions, Proc. Amer. Math. Soc., 24(1970), 312--318.
A. W. Goodman, Univalent functions, Vol. I,II, Mariner Publ Comp. Tampa, Florida, 1983.
W. Janowski, Some extremal problems for certain families of analytic functions I, Ann. Polon. Math., 28(1973), 297--326.
I. B. Jung, Y. C. Kim and H. M. Srivastava, The hardy space of analytic functions associated with certain one-parameter family of integral operator, J. Math. Anal. Appl., 176(1993), 138--147.
S. Kanas and J. Kowalczyk, A note on Briot-Bouquet-Bernoulli differential subordination, Comment. Math. Univ. Carolin., 46(2)(2005), 339--347.
S. Kanas and D. K.-Smket, Harmonic mappings related to functions with bounded boundary rotation and norm of the pre-Schwarzian derivative, Bull. Korean Math. Soc., 51(3)(2014), 803--812.
R. J. Libera, Some classes of regular univalent functions, Proc. Amer. Math. Soc., 16(1965), 755--758.
S. S. Miller and P. T. Mocanu, Differential subordination theory and applications, M. Dekker Inc., N.Y., 2000.
K. I. Noor, On a generalization of $alpha-$, J. Inequal. Pure Appl. Math., 8(16) (2007), 4 pp.
K. I. Noor and S. Z. H. Bukhari, On analytic functions related with generalized Robertson functions, Appl. Math. Comput.,215(2009), 2965-2970.
K. I. Noor and N. E. Cho, Some convolution properties of certain classes of analytic functions, Appl. Math. Lett., 21(11)(2008), 1155--1160.
K. I. Noor and A. Muhammad, On analytic functions with generalized bounded Mocanu variation, Appl. Math. Comput., 196(2008), 802--811.
K. I. Noor, Applications of certain operators to the classes related with generalized Janowski functions, Integral Transforms Spec. Funct., 21(8)(2010), 557--567.
K. I. Noor and S. Hussain, On certain analytic functions associated with Ruscheweyh derivatives and bounded Mocanu variation, J. Math. Anal. Appl., 340(2008), 1145--1152.
S. Owa, Properties of certain integral operators, G.Math. J., 2(5)(1995), 535--545.
S. Owa and H. M. Srivastava, Some applications of the generalized Libera operator, Proc. Japan Acad., 62(1986), 125--128.
K. Padmanabhan and R. Parvatham, Properties of a class of functions with bounded boundary rotation, Ann. Polon. Math., 31(1975), 311--323.
D. Z. Pashkouleva, The starlikeness and spiral-convexity of certain subclasses of analytic functions in Current Topics in Analytic Function Theory, 266--273, World Sci., Singapore, 1992.
J. Patel and P. Sahoo, Some applications of differential subordination to certain one parameter families of integral operators, Indian J. Pure Appl. Math., 35(10)(2004), 1167--1177.
B. Pinchuk, Functions with bounded boundary rotation,Israel J. Math., 10(1971), 7--16.
Y. Polatouglu, M. Bolcal, A. Sen and E. Yavuz, A study on the generalization of Janowski functions in the unit disk,Acta. Math. Acad. Paedag yiregyhaziensis., 22(2006), 27--31.
H. M. Srivastava and S. Owa, A certain one-parameter additive family of operators defined on analytic functions, J. Math. Anal. Appl., 118(1986), 80--87.
J. Stankiewics and Z. Stankiewics, Some applications of the Hadamard convolution in the theory of functions, Ann. Univ. Mariae Curie-Skl odowska Sect. A, 40(1986), 251--265.