22,120 research outputs found

    A new way to prove L'Hospital Monotone Rules with applications

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    Let βˆ’βˆžβ‰€a<bβ‰€βˆž-\infty \leq a<b\leq \infty . Let ff and gg be differentiable functions on (a,b)(a,b) and let gβ€²β‰ 0g^{\prime }\neq 0 on (a,b)(a,b). By introducing an auxiliary function Hf,g:=(fβ€²/gβ€²)gβˆ’fH_{f,g}:=\left( f^{\prime }/g^{\prime }\right) g-f, we easily prove L'Hoipital rules for monotonicity. This offer a natural and concise way so that those rules are easier to be understood. Using our L'Hospital Piecewise Monotone Rules (for short, LPMR), we establish three new sharp inequalities for hyperbolic and trigonometric functions as well as bivariate means, which supplement certain known results.Comment: 19 page

    The monotonicity and convexity of a function involving digamma one and their applications

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    Let L(x,a)\mathcal{L}(x,a) be defined on (βˆ’1,∞)Γ—(4/15,∞)\left( -1,\infty \right) \times \left( 4/15,\infty \right) or (0,∞)Γ—(1/15,∞)\left( 0,\infty \right) \times \left( 1/15,\infty \right) by the formula% \begin{equation*} \mathcal{L}(x,a)=\tfrac{1}{90a^{2}+2}\ln \left( x^{2}+x+\tfrac{3a+1}{3}% \right) +\tfrac{45a^{2}}{90a^{2}+2}\ln \left( x^{2}+x+\allowbreak \tfrac{% 15a-1}{45a}\right) . \end{equation*} We investigate the monotonicity and convexity of the function xβ†’Fa(x)=ψ(x+1)βˆ’L(x,a)x\rightarrow F_{a}\left( x\right) =\psi \left( x+1\right) -\mathcal{L}(x,a), where ψ\psi denotes the Psi function. And, we determine the best parameter aa such that the inequality \psi \left( x+1\right) \right) \mathcal{L}% (x,a) holds for x∈(βˆ’1,∞)x\in \left( -1,\infty \right) or (0,∞)\left( 0,\infty \right) , and then, some new and very high accurate sharp bounds for pis function and harmonic numbers are presented. As applications, we construct a sequence (ln(a))\left( l_{n}\left( a\right) \right) defined by ln(a)=Hnβˆ’L(n,a)l_{n}\left( a\right) =H_{n}-\mathcal{L}\left( n,a\right) , which gives extremely accurate values for Ξ³\gamma .Comment: 20 page
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