Some Simpson type integral inequalities for functions whose third derivatives are (α, m)- GA-convex functions

AbstractBy using power-mean integral inequality and Hölder’s integral inequality, this paper establishes some new inequalities of Simpson type for functions whose three derivatives in absolute value are the class of (α, m)-geometric-arithmetically-convex functions. Finally, some applications to special means of positive real numbers have also been presented

Improved Quantum Artificial Fish Algorithm Application to Distributed Network Considering Distributed Generation

An improved quantum artificial fish swarm algorithm (IQAFSA) for solving distributed network programming considering distributed generation is proposed in this work. The IQAFSA based on quantum computing which has exponential acceleration for heuristic algorithm uses quantum bits to code artificial fish and quantum revolving gate, preying behavior, and following behavior and variation of quantum artificial fish to update the artificial fish for searching for optimal value. Then, we apply the proposed new algorithm, the quantum artificial fish swarm algorithm (QAFSA), the basic artificial fish swarm algorithm (BAFSA), and the global edition artificial fish swarm algorithm (GAFSA) to the simulation experiments for some typical test functions, respectively. The simulation results demonstrate that the proposed algorithm can escape from the local extremum effectively and has higher convergence speed and better accuracy. Finally, applying IQAFSA to distributed network problems and the simulation results for 33-bus radial distribution network system show that IQAFSA can get the minimum power loss after comparing with BAFSA, GAFSA, and QAFSA

k-fractional integral trapezium-like inequalities through ( h , m ) (h,m)(h,m) -convex and ( α , m ) (α,m)(\alpha,m) -convex mappings

Abstract In this paper, a new general identity for differentiable mappings via k-fractional integrals is derived. By using the concept of ( h , m ) $(h,m)$ -convexity, ( α , m ) $(\alpha,m)$ -convexity and the obtained equation, some new trapezium-like integral inequalities are established. The results presented provide extensions of those given in earlier works

Optimality and Duality with Respect to <i>b</i>-(<i>ℰ</i>,<i>m</i>)-Convex Programming

Noticing that E -convexity, m-convexity and b-invexity have similar structures in their definitions, there are some possibilities to treat these three class of mappings uniformly. For this purpose, the definitions of the ( E , m ) -convex sets and the b- ( E , m ) -convex mappings are introduced. The properties concerning operations that preserve the ( E , m ) -convexity of the proposed mappings are derived. The unconstrained and inequality constrained b- ( E , m ) -convex programming are considered, where the sufficient conditions of optimality are developed and the uniqueness of the solution to the b- ( E , m ) -convex programming are investigated. Furthermore, the sufficient optimality conditions and the Fritz&#8315;John necessary optimality criteria for nonlinear multi-objective b- ( E , m ) -convex programming are established. The Wolfe-type symmetric duality theorems under the b- ( E , m ) -convexity, including weak and strong symmetric duality theorems, are also presented. Finally, we construct two examples in detail to show how the obtained results can be used in b- ( E , m ) -convex programming

Properties and Riemann-Liouville fractional Hermite-Hadamard inequalities for the generalized ( α , m ) (α,m)(\alpha,m) -preinvex functions

Abstract The authors first introduce the concepts of generalized ( α , m ) $(\alpha,m)$ -preinvex function, generalized quasi m-preinvex function and explicitly ( α , m ) $(\alpha, m)$ -preinvex function, and then provide some interesting properties for the newly introduced functions. The more important point is that we give a necessary and sufficient condition respecting the relationship between the generalized ( α , m ) $(\alpha, m)$ -preinvex function and the generalized quasi m-preinvex function. Second, a new Riemann-Liouville fractional integral identity involving twice differentiable function on m-invex is found. By using this identity, we establish the right-sided new Hermite-Hadamard-type inequalities via Riemann-Liouville fractional integrals for generalized ( α , m ) $(\alpha,m)$ -preinvex mappings. These inequalities can be viewed as generalization of several previously known results

On Extended General Mittag–Leffler Functions and Certain Inequalities

In this paper, we introduce and investigate generalized fractional integral operators containing the new generalized Mittag&ndash;Leffler function of two variables. We establish several new refinements of Hermite&ndash;Hadamard-like inequalities via co-ordinated convex functions