In this paper, we deal with the following double phase problem $$\begin{aligned} \left\{ \begin{array}{lll} -\text{ div }\left( |\nabla u|^{p-2}\nabla u+a(x)|\nabla u|^{q-2}\nabla u\right) =&{} \gamma \left( \displaystyle \frac{|u|^{p-2}u}{|x|^p}+a(x)\displaystyle \frac{|u|^{q-2}u}{|x|^q}\right) \\ &{}+f(x,u) &{} \text{ in } \Omega ,\\ u=0&{} \partial , \end{array} \right. \end{aligned}$$ where...

For an algebraic number α $\alpha$ and γ ∈ R $\gamma \in \mathbb {R}$ , let be the house, h ( ) $h(\alpha )$ (logarithmic) Weil height, = deg $h_\gamma (\alpha )=(\deg \alpha )^\gamma h(\alpha $\gamma$ -weighted height of . Let f : Q ¯ → [ 0 ∞ $f:\overline{\mathbb {Q}}\rightarrow [0,\infty a function on numbers $\overline{\mathbb {Q}}$ S ⊂ $S\subset \overline{\mathbb The Northcott N $\mathcal {...

In the survey paper, along one of several main lines of bounding the ratio of two gamma functions, the authors retrospect and analyse Wendel’s double inequality, Kazarinoff’s refinement of Wallis’ formula, Watson’s monotonicity, Gautschi’s double inequality, Kershaw’s first double inequality, and the (logarithmically) complete monotonicity results of functions involving ratios of two gamma or q...

In this article, we investigate partial integrals and derivatives of bivariate fractal interpolation functions. We prove also that the mixed Riemann-Liouville fractional integral derivative order $\gamma = (p, q); p > 0,q 0$, functions are again corresponding to some iterated function system (IFS). Furthermore, discuss transforms

We give an infinite family of functions involving the gamma function whose logarithmic derivatives are completely monotonic. Each such function gives rise to an infinitely divisible probability distribution. Other similar results are also obtained for specific combinations of the gamma and q-gamma functions.