By making use of the concept of -calculus, various types of generalized starlike functions of order were introduced and studied from different viewpoints. In this paper, we investigate the relation between various former types of -starlike functions of order . We also introduce and study a new subclass of -starlike functions of order . Moreover, we give some properties of those -starlike functions with negative coefficient including the radius of univalency and starlikeness. Some illustrative examples are provided to verify the theoretical results in case of negative coefficient functions class.
By making use of the concept of fractionalq-calculus, we firstly defineq-extension of the generalization of the generalized Al-Oboudi differential operator. Then, we introduce new class ofq-analogue ofp-valently closed-to-convex function, and, consequently, new class by means of this new general differential operator. Our main purpose is to determine the general properties on such class and geometric properties for functions belonging to this class with negative coefficient. Further, theq-extension of interesting properties, such as distortion inequalities, inclusion relations, extreme points, radii of generalized starlikeness, convexity and close-to-convexity, quasi-Hadamard properties, and invariant properties, is obtained. Finally, we briefly indicate the relevant connections of our presented results to the former results.
Thailand. Its lower section (“tin chok”) is a woven fabric comprising frieze patterns. Harmonised Shape Grammar (HSG) can assist in the design process to achieve harmonized and meaningful designs. We developed the generic framework of HSG to suit a specific context, so that the pattern sets in “tin chok” can be regenerated. We applied this framework to six pattern sets decoded from vintage skirts. This framework consists of five levels of analysis (identifying shape elements, identifying fundamental units, identifying/ understanding a set of rules, generating frieze patterns, arranging frieze patterns on the design structure). We find that their shape elements can be classified into trilateral shapes and quadrilateral shapes. There are 24 fundamental units confined to rectangular spaces. “Tin chok” is designed to have four parts: a main part, two supplementary parts and a hem. By applying the rules of seven frieze groups, a fundamental unit for each part can be generated to seven design derivations. There are 28 possibilities for design derivations per set. However, only 12 possibilities for design derivations appear. Among these, only three out of seven frieze groups are employed.
We introduce a new form of generalized integral operator defined on the class of analytic functionsA0. By making use of this novel integral operator, we give the convexity of other integral operators. We also briefly indicate the relevant connections of our presented results to the formerly reported results. Furthermore, other interesting properties are also discussed.
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