Monotonicity of sequences involving convex and concave functions

Monotonicity of Sequences Involving Convex and Concave Functions

Let f be an increasing and convex (concave) function on [0, 1) and φ a positive increasing concave function on [0,∞) such that φ(0) = 0 and the sequence { φ(i+1) ( φ(i+1) φ(i) − 1 )} i∈N decreases ( the sequence { φ(i) ( φ(i) φ(i+1) − 1 )} i∈N increases ) . Then the sequence { 1 φ(n) ∑n−1 i=0 f ( φ(i) φ(n) )} n∈N is increasing. Mathematics subject classification (2000): 26D15, 26A51.

The Solvability of Concave-Convex Quasilinear Elliptic Systems Involving $p$-Laplacian and Critical Sobolev Exponent

In this work, we study the existence of non-trivial multiple solutions for a class of quasilinear elliptic systems equipped with concave-convex nonlinearities and critical growth terms in bounded domains. By using the variational method, especially Nehari manifold and Palais-Smale condition, we prove the existence and multiplicity results of positive solutions.

Convex and concave relaxations of implicit functions

Short Proofs of the Separation Theorems for L-convex/concave and M-convex/concave Functions

Recently K. Murota has introduced concepts of L-convex function and Mconvex function as generalizations of those of submodular function and base polyhedron, respectively, and has shown separation theorems for L-convex/concave functions and for M-convex/concave functions. The present note gives short alternative proofs of the separation theorems by relating them to the ordinary separation theore...

Monotonicity of some functions involving the gamma and psi functions

Let L(x) := x − Γ(x+t) Γ(x+s) xs−t+1, where Γ(x) is Euler’s gamma function. We determine conditions for the numbers s, t so that the function Φ(x) := − Γ(x+t) xt−s−1 L′′(x) is strongly completely monotonic on (0, ∞). Through this result we obtain some inequalities involving the ratio of gamma functions and provide some applications in the context of trigonometric sum estimation. We also give se...