Univalence and convexity conditions for certain integral operators associated with the Lommel function of the first kind

Ji Hyang Park; Hari Mohan Srivastava; Nak Eun Cho; Ji Hyang Park; Hari Mohan Srivastava; Nak Eun Cho

doi:10.3934/math.2021660

AIMS Mathematics

2021, Volume 6, Issue 10: 11380-11402. doi: 10.3934/math.2021660

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Univalence and convexity conditions for certain integral operators associated with the Lommel function of the first kind

1.
Department of Applied Mathematics, Pukyong National University, Busan 48513, Republic of Korea
2.
Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada
3.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
4.
Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, AZ1007 Baku, Azerbaijan
5.
Section of Mathematics, International Telematic University Uninettuno, I-00186 Rome, Italy

Received: 17 May 2021 Accepted: 29 July 2021 Published: 06 August 2021
MSC : 30C45, 33C10

A useful family of integral operators and special functions plays a crucial role on the study of mathematical and applied sciences. The purpose of the present paper is to give sufficient conditions for the families of integral operators, which involve the normalized forms of the generalized Lommel functions of the first kind to be univalent in the open unit disk. Furthermore, we determine the order of the convexity of the families of integral operators. In order to prove main results, we use differential inequalities for the Lommel functions of the first kind together with some known properties in connection with the integral operators which we have considered in this paper. We also indicate the connections of the results presented here with those in several earlier works on the subject of our investigation. Moreover, some graphical illustrations are provided in support of the results proved in this paper.
- analytic functions,
- univalent functions,
- integral operators,
- special functions,
- Lommel functions,
- univalence,
- convexity,
- quantum (or $ q $-) calculus
Citation: Ji Hyang Park, Hari Mohan Srivastava, Nak Eun Cho. Univalence and convexity conditions for certain integral operators associated with the Lommel function of the first kind[J]. AIMS Mathematics, 2021, 6(10): 11380-11402. doi: 10.3934/math.2021660

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Abstract

A useful family of integral operators and special functions plays a crucial role on the study of mathematical and applied sciences. The purpose of the present paper is to give sufficient conditions for the families of integral operators, which involve the normalized forms of the generalized Lommel functions of the first kind to be univalent in the open unit disk. Furthermore, we determine the order of the convexity of the families of integral operators. In order to prove main results, we use differential inequalities for the Lommel functions of the first kind together with some known properties in connection with the integral operators which we have considered in this paper. We also indicate the connections of the results presented here with those in several earlier works on the subject of our investigation. Moreover, some graphical illustrations are provided in support of the results proved in this paper.

References

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