Volume 36, Issue 4

We introduce a sequence of real functions which converges to the constant function x↦γ${x\mapsto\gamma}$ where γ is the Euler–Mascheroni constant. We determine a point where the resulting approximation of γ is very fast. Furthermore, we use our approach to improve known estimates in the pursuit of a conjecture related to harmonic numbers, part of which is settled.

A class of sums of the type ∑n=1,n≡a1,…,ar⁢(mod⁡m)∞1n2⁢k$\sum^{\infty}_{n=1,n\equiv a_{1},\ldots,a_{r}\,(\operatorname{mod}m)}\frac{1}{% n^{2k}}$ is evaluated, where k , m and r are positive integers with m≥2$m\geq 2$ and a1,…,ar$a_{1},\ldots,a_{r}$ are integers satisfying 1≤a1<a2<⋯<ar≤m-1$1\leq a_{1}<a_{2}<\cdots<a_{r}\leq m-1$.

In this paper, we present some new Lyapunov and Hartman type inequalities for second-order equations with mixed nonlinearities: x′′⁢(t)+p⁢(t)⁢|x⁢(t)|β-1⁢x⁢(t)+q⁢(t)⁢|x⁢(t)|γ-1⁢x⁢(t)=0,$x^{\prime\prime}(t)+p(t)|x(t)|^{\beta-1}x(t)+q(t)|x(t)|^{\gamma-1}x(t)=0,$ where p⁢(t)${p(t)}$, q⁢(t)${q(t)}$ are real-valued functions and 0<γ<1<β<2${0<\gamma<1<\beta<2}$. No sign restrictions are imposed on the potential functions p⁢(t)${p(t)}$ and q⁢(t)${q(t)}$. The inequalities obtained generalize the existing results for the special cases of this equation in the literature.

In this paper we introduce new refinements of both the discrete and the classical Jensen’s inequality. First, we give the weighted version of a recent cyclic refinement. By using this result, we obtain new refinements of the classical Jensen’s inequality. We investigate m -exponential convexity of some functionals coming from the new refinements. To apply our results we define some new mixed symmetric means, generalized means, and Cauchy means, and study their properties.

In 1993 one of the authors formulated some conjectures on monotonicity of ratios for exponential series sections. These lead to a more general conjecture on monotonicity of ratios of Kummer hypergeometric functions, which remained open ever since. In this paper we prove some conjectures for Kummer hypergeometric functions and its further generalizations for Gauss and generalized hypergeometric functions. The results are also closely connected with Turán-type inequalities.

This paper deals with the time-dependent Stokes problem with Navier boundary conditions on Lp${L^{p}}$-spaces. We prove the analyticity of the Stokes semigroup with Navier boundary conditions on some spaces to be determined. This analyticity allows us to solve the evolutionary Stokes problem and to obtain weak, strong and very weak solutions.

Let (M,g)${(M,g)}$ be a smooth compact Riemannian manifold of dimension n≥3${n\geq 3}$. We fix a point x0∈M${x_{0}\in M}$ and s∈(0,2)${s\in(0,2)}$. We investigate a sufficient condition for the existence of a distributional continuous positive solution for the Hardy–Sobolev equation Δg⁢u+a⁢(x)⁢u=u2⋆⁢(s)-1dg⁢(x,x0)s+h⁢uq-1 in ⁢M,$\Delta_{g}u+a(x)u=\frac{u^{2^{\star}(s)-1}}{d_{g}(x,x_{0})^{s}}+hu^{q-1}\quad% \text{in }M,$ where Δg:=-divg⁡(∇)${\Delta_{g}:=-\operatorname{div}_{g}(\nabla)}$ is the Laplace–Beltrami operator, a,h∈C0⁢(M)${a,h\in C^{0}(M)}$, h≥0${h\geq 0}$, dg${d_{g}}$ is the Riemannian distance on (M,g)${(M,g)}$, 2⋆⁢(s)=2⁢(n-s)n-2${2^{\star}(s)=\frac{2(n-s)}{n-2}}$, and q∈(2,2⋆)${q\in(2,2^{\star})}$ with 2⋆=2⋆⁢(0)${2^{\star}=2^{\star}(0)}$. We prove that the existence of a mountain pass solution for the above perturbative equation depends only on the perturbation when n≥4${n\geq 4}$, while for n=3${n=3}$, it depends on other conditions involving the perturbation and the global geometry of the manifold.