On Opial-Traple type inequalities for β-partial derivatives

Abstract: In the paper, we introduce a new partial derivative call it β-partial derivatives as the most natural extensions of the limit definitions of the partial derivative and the β-derivative, which obeys classical properties including: continuity, linearity, product rule, quotient rule, power rule, chain rule and vanishing derivatives for constant functions. As applications, we establish some new Opial-Traple type inequalities for the β-partial derivatives.

In the paper, we give a new concept of β-partial derivatives as the most natural extension of the familiar limit definition of the partial derivative. We show also that the β-partial derivatives obeys classical properties including: continuity, linearity, product rule, quotient rule, power rule, chain rule and vanishing derivatives for constant functions. As applications, we establish some new Opial-Traple type inequalities for the β-partial derivatives.

The β-partial derivatives
There exist a quite few definitions of fractional derivatives in the literatures, we will present one definition. Given a function f : [0, ∞] → R. Then for all β ∈ (0, 1] and x ∈ (0, ∞), the β-derivative, defined by (see [18] provided the limits exist, where Γ(·) is the usual Γ function. A function f is β-differentiable at a point x ≥ 0, if the limits in (2.1) exist and are finite.
In this section, we give a new definition as the most natural extensions of the limit definitions of the partial derivative and the β-derivative. To this end, we start with the following definition which is a generalization of the classical partial derivative and β-derivative, respectively.
for all x ≥ a and β ∈ (0, 1]. If the limit of the above exists, then f (x, y) is said to be β-partial x differentiable and call A a P β x ( f (x, y)) as β-partial x derivatives of f (x, y). (ii) the beta-partial y derivative of a function f (x, y) is defined as for all y ≥ b and β ∈ (0, 1]. If the limit of the above exists, then f (x, y) is said to be β-partial y differentiable and call A b P β y ( f (x, y)) as β-partial y derivatives of f (x, y). β-partial x, and β-partial y differentiable are collectively called β-partial differentiable. Remark 2.2 Putting β = 1 and a = b = 0 in (2.2) and (2.3), the β-partial derivatives A a P 1 x ( f (x, y)) and A b P 1 y ( f (x, y)) just are the usual partial derivatives ∂ f (x,y) ∂x and ∂ f (x,y) ∂y , respectively. Let f (x, y) become f (x) and with a proper transformation in (2.2), and let a = 0, the β-partial x derivatives A a P β x ( f (x, y)) reduces to the well-known β-derivatives A 0 D β x ( f (x)).

Properties for β-partial derivatives
In this section, we give several results for the β-partial derivatives such as the continuity, linearity, product rule, quotient rule, power rule, chain rule and vanishing derivatives for constant functions. and This completes the proof. Theorem 3.2 Assuming that f (x, y) and g(x, y) are two β-partial x differentiable functions with β ∈ (0, 1], then the following relations can be satisfied: y)), for all a and b real number.
(v) A a P β x (λ) = 0 for λ any given constant. Proof Obviously, the (i) and (v) follow immediately from Definition 2.1. Let Since f (x, y) is β-partial x differentiable at x ≥ a, and by using L'Hospital rule, we obtain This completes the proof of (ii). On the other hand, from (ii), we have A a P β x ( f (x, y) · g(x, y)) = , y)).
This completes the proof of (iii). The proof of the (iv) is similar to (iii). Here, we omit this details. This completes the proof. Theorem 3.3 Assuming that f (x, y) and g(x, y) are two β-partial y differentiable functions with β ∈ (0, 1], then the following relations can be satisfied: y)), for all a and b real number.

3)
where f g (g(x, y)) denotes the derivative of function f to g(x, y).
This completes the proof. This chain rule theorem is important, but it is also understood. In order for the reader to better understand this theorem, we give another proof below. Second proof Let Obviously, if ε → 0, then δ → 0. From the hypotheses, we obtain , y)).
This completes the proof. Let f (x, y) and g(x, y) change f (x) and g(x) with a proper transformation in Theorem 3.4, it becomes the following result, which was established in [18]. Corollary 3.5 Let f (x) : [0, ∞) → R be a function such that f (x) is β-differentiable. If g(x, y) is a function defined in the range of f (x, y) and also differentiable, then where, A 0 D β x ( f (x)) denotes the β-derivatives of f (x). Theorem 3.6 Let f (x, y) : [a, ∞) × [b, ∞) → R be a function such that f (x, y) is β-partial y differentiable. If g(x, y) is a function defined in the range of f (x, y) and also β-partial y differentiable, then where f g (g(x, y)) denotes the derivative of function f to g(x, y). Proof This follows immediately from the proof of Theorem 3.4 with a proper transformation.

Opial-Traple type inequalities for β-partial derivatives
In the section, we establish Opial-Traple type inequalities for the β-partial derivatives.
This completes the proof.
where A 0 D β s is as in (2.1), and Proof Let u(s, t) and p(s, t) change to u(s) and p(s), respectively, and with a proper transformation, and let a = 0, (4.6) follows immediately from (4.2). Proof This follows immediately from Theorem 4.2 with β = 1. Let p(s, t) and u(s, t) reduce to p(t) and u(t), respectively, and with suitable modifications, and let a = 0 and b = h, (4.7) becomes the following result. This is just an inequality which was established in [14]. Here, we call it Opial-Traple's inequality.