In this paper, we establish some lower bound estimates for a certain class of rational functions on a disk with prescribed poles and restricted zeros. The results obtained strengthen some known results for rational functions and in turn produce generalizations of some polynomial inequalities as well.
In this manuscript, we present a collection of inequalities that extend and generalize the important results of the findings from [14], with the latter serving as specific cases within our study. Our research takes many forms of fractional differential inequalities, including the comprehensive structure of the Caputo fractional derivative operator with respect to a function . Furthermore, we demonstrate the practical applications of a--convex functions in the domain of fractional calculus within this framework.
In this paper, singularly perturbed time-fractional parabolic reactiondiffusion of initial boundary value problem is studied. The time-fractional derivative is applied in the Caputo fractional sense and handled by implicit Euler method. The spatial derivative is approximated by fitted cubic B-spline collocation method on a uniform mesh. Convergence analysis of the scheme is conducted and it is accurate of order O(h(2) + (triangle t(2-alpha))). To test the effectiveness of proposed method two model examples are considered. The results from the experiment confirm that the scheme is uniformly convergent and has twin layers at the end spatial domain.
We obtain the celebrated Hardy's inequality in the context of variable exponent sequence spaces.
A new geometric background of graph invariants was introduced by Gutman, of which the simplest is the second Somb or index SO2, defined as SO2 = SO2(G) =& sum;(uv is an element of E)|d(2)G(u)-d(2)G(v)| / d(2)G(u)+d(2)G(v), where G = (V, E) is a simple graph and d(G)(v) denotes the degree of v in G. In this paper, the chemical applicability of the second Somb or index is investigated and it is shown that the second Somb or index is useful in predicting physicochemical properties with high accuracy compared to some well-established and often used indices. Also, we obtain a bound for the second Somb or index among all (molecular) trees with fixed numbers of vertices, and characterize those molecular trees achieving the extremal value.
