Kyungpook Mathematical Journal 2020; 60(1): 126-132
Published online March 31, 2020
Copyright © Kyungpook Mathematical Journal.
Hardy Spaces of Certain Convolution Operator
Rajbala and Jugal Kishore Prajapat∗
Received: January 14, 2018; Revised: March 20, 2019; Accepted: April 9, 2019
In this article, we determine sufficient conditions on the parameters of a generalized convolution operator to ensure that it belongs to the Hardy space and to the space of bounded analytic functions. We exhibit the utility of these results by deducing several interesting examples.
Keywords: analytic functions, bounded analytic functions, Hardy spaces.
Let ℋ denote the class of analytic functions in the open unit disk , and be the subclass of ℋ consisting of normalized analytic functions of the form
For the analytic functions
Further, for functions , given by
Let ℋ∞ denote the space of all bounded analytic functions in . This is Banach algebra with respect to the norm . For
(see [3, p. 2]). For 1 ≤
In , Ponnusamy studied the Hardy space of hypergeometric functions. Further, Baricz  obtained the conditions for the generalized Bessel functions such that it belongs to Hardy space and Yagmur and Orhan  studied the same problem for the generalized Struve functions.
where the function
Corresponding to fractional integral operator
In order to derive our main results, we recall here the following lemmas:
Our first main result is given by Theorem 2.3 below.
From the definition of operator
which shows that 0 <
Now, we shall show that lim
This implies that the series in
Also, using the known result [3, Theorem 3.11],
If we consider
which in view of
The authors wish to thank the referee for her/his valuable suggestions, which have improved the paper. The work was supported by the University Grant Commission, India (Project Number: MRP-MAJOR-MATH-2013-19114).
- Á. Baricz.
Bessel transforms and Hardy space of generalized Bessel functions. Mathematica., 48(71)(2006), 127-136.
- BC. Carlson, and DB. Shaffer.
Starlike and prestarlike hypergeometric functions. SIAM J Math Anal., 15(1984), 737-745.
- PL. Duren.
Theory of ℋp space. Pure and Applied Mathematics, 38, Academics Press, New York/London, 1970.
- A. Erdeilyi, W. Magnus, F. Oberhettinger, and FG. Tricomi. Higher transcendental functions, I,
, McGraw–Hill Book Company, New York, 1953.
- J. Hadamard.
Essai sur l’etude des fonctions donnees par leur developpement de Taylor. J Math Pures Appl., 8(1892), 101-186.
- RW. Ibrahim.
On generalized Srivastva-Owa fractional operators in the unit disk. Adv Difference Equ., 2011(2011) 10 pp, 55.
- RW. Ibrahim.
Studies on generalized fractional operators in complex domain. Mathematics Without Boundaries, , Springer, New York, 2014:273-284.
- Y. Komatu.
On a one-parameter additive family of operators defined on analytic functions regular in the unit disk. Bull Fac Sci Engrg Chou Univ., 22(1979), 1-22.
- RJ. Libera.
Some classes of regular univalent functions. Proc Amer Math Soc., 16(1965), 755-758.
- S. Ponnusamy.
The Hardy spaces of hypergeometric functions. Complex Var Theory Appl., 29(1)(1996), 83-96.
- JK. Prajapat, RK. Raina, and J. Sokol.
Dependence conditions for analytic functions under fractional differintegral operators. Math Sci Res J., 15(12)(2011), 333-340.
- A. Prudnikov, Yu. Brychkov, and O. Marchev. Integral and series, Vol 3 More special functions,
, Gordon & Breach Science Publishers, New York, 1990.
- HM. Srivastva, and S. Owa. Univalent functions, fractional calculus and their applications,
, John Wiley & Sons, New York, 1989.
- J. Stankiewicz, and Z. Stankiewicz.
Some applications of Hadamard convolution in the theory of functions. Ann Univ Mariae Curie-Skłodowska Sect A., 40(1986), 251-265.
- T. Suffridge.
Some remarks on convex maps of the unit disk. Duke Math J., 37(4)(1970), 775-777.
- N. Yağmur, and H. Orhan.
Hardy space of generalized Struve functions. Complex Var Elliptic Equ., 59(7)(2014), 929-936.