Chu Yu-Ming’s research while affiliated with Hunan University and other places

In this paper, we prove that the double inequality \begin{equation*} 1+\alpha r'^2<\frac{\mathcal{K}_{a}(r)}{\sin(\pi a)\log(e^{R(a)/2}/r')}<1+\beta r'^2 \end{equation*} holds for all $a\in (0, 1/2]$ and $r\in (0, 1)$ if and only if $\alpha\leq \pi/[R(a)\sin(\pi a)]-1$ and $\beta\geq a(1-a)$, where $r'=\sqrt{1-r^2}$, $\mathcal{K}_{a}(r)$ is the generalized elliptic integral of the first kind and R(x) is the Ramanujan constant function. Besides, as the key tool, the series expression for the Ramanujan constant function R(x) is given.

In this paper, we discuss the Schur convexity, Schur geometrical convexity and Schur harmonic convexity of the weighted arithmetic integral mean and Chebyshev functional. Several sufficient conditions, and necessary and sufficient conditions are established.

We establish several necessary and sufficient conditions for oscillation of the solutions of the following even-order differential equation x (n) (t)+q(t)x γ (t)=0,niseven, where q∈C([t 0 ,∞),ℝ + ) and γ is the quotient of odd positive integers.

We obtain the analytical general solution of the linear fractional differential equations with constant coefficients by Adomian decomposition method under nonhomogeneous initial value condition, which is in the sense of the Caputo fractional derivative.

We answer the question: for , what are the greatest value and the least value such that the double inequality holds for all with . Here, , , and denote the power of order , Seiffert, and geometric means of two positive numbers and , respectively.

For $p\in\mathbb{R}$, the one-parameter mean $J_{p}(a,b)$, arithmetic mean A(a,b), and harmonic mean H(a,b) of two positive real numbers a and b are defined by\begin{equation*}J_{p}(a,b)=\begin{cases}\tfrac{p(a^{p+1}-b^{p+1})}{(p+1)(a^p-b^p)}, & a\neq b,p\neq 0,-1,\\\tfrac{a-b}{\log{a}-\log{b}}, & a\neq b,p=0,\\\tfrac{ab(\log{a}-\log{b})}{a-b}, & a\neq b,p=-1,\\a, & a=b,\end{cases}\end{equation*}$A(a,b)=\tfrac{a+b}{2}$, and $H(a,b)=\tfrac{2ab}{a+b}$,respectively. In this paper, we answer the question: For $\alpha\in(0,1)$, what are the greatest value $r_{1}$ and the least value $r_{2}$ such that the double inequality \(J_{r_{1}}(a,b)

For 𝑝∈ℝ, the power mean 𝑀𝑝(𝑎,𝑏) of order 𝑝, logarithmic mean 𝐿(𝑎,𝑏), and arithmetic mean 𝐴(𝑎,𝑏) of two positive real values 𝑎 and 𝑏 are defined by 𝑀𝑝(𝑎,𝑏)=((𝑎𝑝+𝑏𝑝)/2)1/𝑝, for 𝑝≠0 and 𝑀𝑝√(𝑎,𝑏)=𝑎𝑏, for 𝑝=0, 𝐿(𝑎,𝑏)=(𝑏−𝑎)/(log𝑏−log𝑎), for 𝑎≠𝑏 and 𝐿(𝑎,𝑏)=𝑎, for 𝑎=𝑏 and 𝐴(𝑎,𝑏)=(𝑎+𝑏)/2, respectively. In this paper, we answer the question: for 𝛼∈(0,1), what are the greatest value 𝑝 and the least value 𝑞, such that the double inequality 𝑀𝑝(𝑎,𝑏)≤𝛼𝐴(𝑎,𝑏)+(1−𝛼)𝐿(𝑎,𝑏)≤𝑀𝑞(𝑎,𝑏) holds for all 𝑎,𝑏>0?

We present a new method to study analytic inequalities. As for its applications, we prove the well-known Hölder inequality and establish several new analytic inequalities.

We answer the question: for α,β,γ∈(0,1) with α+β+γ=1, what are the greatest value p and the least value q, such that the double inequality Lp(a,b)<Aα(a,b)Gβ(a,b)Hγ(a,b)<Lq(a,b) holds for all a,b>0 with a≠b? Here Lp(a,b), A(a,b), G(a,b), and H(a,b) denote the generalized logarithmic, arithmetic, geometric, and harmonic means of two positive numbers a and b, respectively.

We prove that G(x,y)=|∫yxf(t)dt| is multiplicatively concave on [a,b]×[a,b] if f:[a,b]⊂(0,∞)→(0,∞) is continuous and multiplicatively concave.