(untitled) (Contemporary Mathematics, 2024)
The classical theories of differential superordination and subordination have been extended to strong differential superordination and respectively, strong differential subordination. The two new theories have progressed well, revealing significant findings when various operators and specific hypergeometric functions have been included in the studies. The research revealed by this work expands the topic of the investigation by incorporating aspects of fractional calculus and quantum calculus. An extended version of q-hypergeometric function is introduced to correspond to the study of functions from the classes that were previously described and that are particularly defined for strong differential superordination and subordination theories. This work defines the Riemann-Liouville fractional integral applied to the extended q-hypergeometric function, used to get strong differential subordinations and superordination results. The theorems established for the strong differential superordination and subordination, establish the best subordinants and respectively the best dominants. Interesting corollaries are exposed for certain functions regarded as best subordinant or best dominant due to their particular geometric characteristics. Sandwich-type theorems and consequences conclude the study, stated to connect the outcomes obtained by applying the dual theories.