On integral representations of (α,β,γ)-type and (α,β,γ)-weak type of meromorphic function

Bhattacharyya, Sarmila; Biswas, Tanmay; Biswas, Chinmay

doi:10.21608/jfca.2023.241241.1042

On integral representations of (α,β,γ)-type and (α,β,γ)-weak type of meromorphic function

Document Type : Regular research papers

Authors

¹ Department of Mathematics, Netaji Mahavidyalaya, P.O.- Arambagh, Dist.-Hooghly, PIN-712601, West Bengal, India

² Rajbari, Rabindrapally, R. N. Tagore Road, P.O. Krishnagar, P.S.- Katwali, Dist-Nadia, PIN- 741101, West Bengal, India.

³ Department of Mathematics, Nabadwip Vidyasagar College, Nabadwip, Dist.-Nadia, PIN-741302, West Bengal, India

10.21608/jfca.2023.241241.1042

Abstract

In complex analysis, order and type are classical growth indicators of entire and meromorphic functions. During the past decades, several authors have made the close investigations on the properties of entire and meromorphic functions in different directions using the concepts of order, the iterated p-order [8, 11], the (p,q)-th order [6, 7], (p,q)-ϕ order [10] and achieved many valuable results. But in [3], Chyzhykov et al. showed that both definitions of iterated p-order and the (p,q)-th order have the disadvantage that they do not cover arbitrary growth (see [3], Example 1.4). They used more general scale, called the ϕ-order (see [3]). On the other hand, Heittokangas et al. [4] have introduced another new concept of ϕ-order of entire and meromorphic functions considering ϕ as subadditive function. Considering all these aspects, Belaïdi et al. [1, 2] have extended the above ideas and have introduced the definitions of (α,β,γ)-order and (α,β,γ)-type of entire and meromorphic functions. In this paper, we establish the integral representations of (α,β,γ)-type and (α,β,γ)-weak type of a meromorphic function. We also investigate their equivalence relation under some certain conditions.

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