The operator defined as the fractional integral of confluent hypergeometric function was introduced and studied in previously written papers in view of the classical theory of differential subordination. In this paper, the same operator is studied using concepts from the theory of fuzzy differential subordination and superordination. The original theorems contain fuzzy differential subordinations and superordinations for which the fuzzy best dominant and fuzzy best subordinant are given, respectively. Interesting corollaries are obtained for particular choices of the functions acting as fuzzy best dominant and fuzzy best subordinant. A nice sandwich-type theorem is stated combining the results given in two theorems proven in this paper using the two dual theories of fuzzy differential subordination and fuzzy differential superordination.
The fractional integral of confluent hypergeometric function is used in this paper for obtaining new applications using concepts from the theory of fuzzy differential subordination and superordination. The aim of the paper is to present new fuzzy differential subordinations and superordinations for which the fuzzy best dominant and fuzzy best subordinant are given, respectively. The original theorems proved in the paper generate interesting corollaries for particular choices of functions acting as fuzzy best dominant and fuzzy best subordinant. Another contribution contained in this paper is the nice sandwich-type theorem combining the results given in two theorems proved here using the two theories of fuzzy differential subordination and fuzzy differential superordination.
In this paper we study a certain differential subordination related to the harmonic mean and its symmetry properties, in the case where a dominant is a linear function. In addition to the known general results for the differential subordinations of the harmonic mean in which the dominant was any convex function, one can study such differential subordinations for the selected convex function. In this case, a reasonable and difficult issue is to look for the best dominant or one that is close to it. This paper is devoted to this issue, in which the dominant is a linear function, and the differential subordination of the harmonic mean is a generalization of the Briot–Bouquet differential subordination.
"Let 1 and 2 belong to a certain class of normalized analytic univalent functions in the open unit disk of the complex plane. Sufficient conditions are obtained for normalized analytic functions to satisfy the double subordination chain 1() ≺() ≺ 2() , then we obtain 1() is the best subordinant, 2() is the best dominant. Also we derive some sandwich –type result.
Functionsf(z)=z+∑2∞anznthat are analytic in the unit disk and satisfy the differential equationf'(z)+αzf''(z)+γz2f'''(z)=g(z)are considered, wheregis subordinated to a normalized convex univalent functionh. These functionsfare given by a double integral operator of the formf(z)=∫01∫01G(ztμsν)t-μs-νds dtwithG'subordinated toh. The best dominant to all solutions of the differential equation is obtained. Starlikeness properties and various sharp estimates of these solutions are investigated for particular cases of the convex functionh.
We solve the problem of finding the largest domainDfor which, under givenψandq, the differential subordinationψ(p(z),zp′(z))∈D⇒p(z)≺q(z), whereDandq(𝒰)are regions bounded by conic sections, is satisfied. The shape of the domainDis described by the shape ofq(𝒰). Also, we find the best dominant of the differential subordinationp(z)+(zp′(z)/(βp(z)+γ))≺pk(z), when the functionpk(k∈[0,∞))maps the unit disk onto a conical domain contained in a right half-plane. Various applications in the theory of univalent functions are also given.
Fuzzy Differential Sandwich Theorems Involving the Fractional Integral of Confluent Hypergeometric Function
