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Y-function and L'Hospital-type Monotonicity Rules with Nabla and Diamond-Alpha Derivatives on Time Scales

Xiao-Yue Du, Zhong-Xuan Mao, Jing-Feng Tian

Abstract

The main objective of this paper is to establish the $Y$-function and L'Hospital-type monotonicity rules with nabla and diamond-alpha derivatives on time scales.

Y-function and L'Hospital-type Monotonicity Rules with Nabla and Diamond-Alpha Derivatives on Time Scales

Abstract

The main objective of this paper is to establish the -function and L'Hospital-type monotonicity rules with nabla and diamond-alpha derivatives on time scales.
Paper Structure (3 sections, 8 theorems, 34 equations)

This paper contains 3 sections, 8 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. ${Y}$-function and L'Hospital-type Monotonicity Rules with Nabla Derivatives on Time Scales
  3. $Y$-function and L'Hospital-type Monotonicity Rules with Diamond-Alpha Derivatives on Time Scales

Key Result

Proposition 1.1

ref1 Let $-\infty \leq a<b\leq \infty$. Let $f$ and $g$ be differentiable functions on $(a,b)$ and let $g^{\prime}\neq 0$ on $(a,b)$. Let $Y_{f,g}$ be defined on $\left(a,b\right)$ by (1.1). Then the following statements are true. (i) The function is even with respect to $g$ and odd with respect to (ii) If $f$ and $g$ are twice differentiable on $(a,b)$, then (iii) If $g\neq 0$ on $\left( a,b\ri

Theorems & Definitions (19)

  • Proposition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Remark 2.1
  • Proposition 2.1
  • Proposition 2.2
  • Remark 2.2
  • proof
  • ...and 9 more