For simplicity, we adopt the following rules: x, x0, y, y0, z, z0, r denote real numbers, u, u0 denote elements of R3, f , f1, f2 denote partial functions from R3 to R, R denotes a rest, and L denotes a linear function. Let f be a partial function from R3 to R and let u be an element of R3. We say that f is partial differentiable on 1st-1st coordinate in u if and only if the condition (Def. 1) ...

In this paper, we define the β-partial derivative as well β-directional of a multi-variable function based on β-difference operator, Dβ, which is defined by Dβf(t)=f(β(t))−f(t)/β(t)−t, where β strictly increasing continuous function. Some properties are proved. Furthermore, β-gradient vector and directional introduced. Finally, deduce Hahn-partial Hahn-directional derivatives associated with Ha...

A continuous complex-valued function F in a domain $$D\subseteq \mathbf {C}$$ is poly-analytic of order $$\alpha $$ if it satisfies $$\partial ^{\alpha }F/\partial \overline{z}^{\alpha }=0$$ . One can show that has the form $$F(z)=\sum _{k=0}^{\alpha -1}\overline{z}^{k}A_{k}(z)$$ , where each $$A_k$$ an analytic function. In this paper, we prove existence Landau constant for functions and ...

The notation and terminology used here are introduced in the following papers: [25], [28], [29], [4], [30], [6], [14], [5], [2], [24], [10], [26], [27], [19], [15], [12], [13], [11], [31], [20], [3], [1], [16], [21], [17], [23], [7], [8], [22], [18], and [9]. For simplicity, we use the following convention: n denotes a natural number, r, s denote real numbers, z denotes a complex number, C1, C2...

We obtain necessary and sufficient conditions for the existence of solutions to boundary-value problem $$ \Delta_p u=f\quad\text{on}\quad M,\qquad |\nabla u|^{p-2}\,\frac {\partial u}{\partial \nu}\bigg|_{\partial M}=h, where $$p > 1$$ is a real number, $$M$$ connected oriented complete Riemannian manifold with boundary, $$\nu$$ outer normal vector $$\partial M$$ .