Advances in Analysis

6

Authors: Ivan Kupka

In [1] we defined a notion of a generalized derivative for functions defined on a general topological space with values in a linear topological space. Here we develop a theory of Taylor series for this generalized setting.

9

Authors: K. R. Karthikeyan, K. Amarender Reddy, M. Thirucheran

We introduce a new subclass of spiralike biunivalent functions involving q-differential operator. We obtain the coefficient estimates and Fekete-Szegö inequalities for the functions belonging to this class. Relevant connections with various other known classes have been established.

18

Authors: Chenkuan Li, Kyle Clarkson, Vrajna Patel

Let {τn} be a certain sequence of functions in D converging to 1 in D 0 . The commutative neutrix convolution f ♦∗ g of two distributions f and g in D 0 is defined to be the neutrix limit of the sequence 1 2 {(fτn) ∗ g + f ∗ (gτn)} , provided the limit exists. We present relations between this new convolution and other existing distributional convolutions, and demonstrate its strong computational power in evaluating convolutions as well as applications to defining new fractional derivatives and integrals of generalized functions in the new space H which contains D 0 (R +). The neutrix convolutions x λ − ♦∗ x µ + for λ, µ, λ + µ 6= 0, ±1, ±2, · · · and x λ − ♦∗ x s + for λ 6= 0, ±1, ±2, · · · and s = 0, 1, 2, · · · are evaluated, from which other neutrix convolutions are deduced.