Application on Fuzzy Third-Order Subordination and Superordination Connected with Lommel Function

Abstract

This work is based on the recently introduced concepts of third-order fuzzy differential subordination and its dual, third-order fuzzy differential superordination. In order to obtain the new results that add to the development of the newly initiated lines of research, a new operator is defined here using the concept of convolution and the normalized Lommel function. The methods focusing on the basic concept of admissible function are employed. Hence, the investigation of new third-order fuzzy differential subordination results starts with the definition of the suitable class of admissible functions. The first theorems discuss third-order fuzzy differential subordinations involving the newly introduced operator. The following result shows the conditions needed such that the fuzzy best dominant can be found for a third-order fuzzy differential subordination. Next, dual results are obtained by employing the methods of third-order fuzzy differential superordination based on the same concept of an admissible function. A suitable class of admissible functions is introduced and new third-order fuzzy differential superordinations are obtained, showing how the best subordinant can be obtained under certain restrictions. As a conclusion of this study, sandwhich-type results are derived, linking the outcome of the two dual fuzzy theories.

Keywords

• fuzzy set
• third-order fuzzy differential subordination
• third-order fuzzy differential superordination
• fuzzy dominant
• fuzzy best dominant
• fuzzy subordinant