In the article, we prove that the double inequalities π e − x 2 ( x + a ) < K 0 ( x ) < π e − x 2 ( x + b ) , 1 + 1 2 ( x + a ) < K 1 ( x ) K 0 ( x ) < 1 + 1 2 ( x + b ) $$ \frac{\sqrt{\pi}e^{-x}}{\sqrt{2(x+a)}}< K_{0}(x)< \frac{\sqrt{\pi }e^{-x}}{\sqrt{2(x+b)}},\qquad 1+ \frac{1}{2(x+a)}< \frac {K_{1}(x)}{K_{0}(x)}< 1+\frac{1}{2(x+b)} $$ hold for all x > 0 $x>0$ if and only if a ≥ 1 / 4 $a\geq1/4$ and b = 0 $b=0$ if a , b ∈ [ 0 , ∞ ) $a, b\in[0, \infty)$ , where K ν ( x ) $K_{\nu}(x)$ is the modified Bessel function of the second kind. As applications, we provide bounds for K n + 1 ( x ) / K n ( x ) $K_{n+1}(x)/K_{n}(x)$ with n ∈ N $n\in\mathbb{N}$ and present the necessary and sufficient condition such that the function x ↦ x + p e x K 0 ( x ) $x\mapsto\sqrt {x+p}e^{x}K_{0}(x)$ is strictly increasing (decreasing) on ( 0 , ∞ ) $(0, \infty)$ .

In the article, we prove that the double inequality x 2 + p 0 x + p 0 < Γ ( x + 1 ) < x 2 + 9 / 5 x + 9 / 5 $$ \frac{x^{2}+p_{0}}{x+p_{0}}< \Gamma(x+1)< \frac{x^{2}+9/5}{x+9/5} $$ holds for all x ∈ ( 0 , 1 ) $x\in(0, 1)$ , we present the best possible constants λ and μ such that λ ( x 2 + 9 / 5 ) x + 9 / 5 ≤ Γ ( x + 1 ) ≤ μ ( x 2 + p 0 ) x + p 0 $$ \frac{\lambda(x^{2}+9/5)}{x+9/5}\leq\Gamma(x+1)\leq\frac{\mu (x^{2}+p_{0})}{x+p_{0}} $$ for all x ∈ ( 0 , 1 ) $x\in(0, 1)$ , and we find the value of x ∗ $x^{\ast}$ in the interval ( 0 , 1 ) $(0, 1)$ such that Γ ( x + 1 ) > ( x 2 + 1 / γ ) / ( x + 1 / γ ) $\Gamma(x+1)>(x^{2}+1/\gamma)/(x+1/\gamma)$ for x ∈ ( 0 , x ∗ ) $x\in(0, x^{\ast})$ and Γ ( x + 1 ) < ( x 2 + 1 / γ ) / ( x + 1 / γ ) $\Gamma(x+1)<(x^{2}+1/\gamma)/(x+1/\gamma )$ for x ∈ ( x ∗ , 1 ) $x\in(x^{\ast}, 1)$ , where Γ ( x ) $\Gamma(x)$ is the classical gamma function, γ = lim n → ∞ ( ∑ k = 1 n 1 / k − log n ) = 0.577 … $\gamma=\lim_{n\rightarrow\infty}(\sum_{k=1}^{n}1/k-\log n)=0.577\ldots$ is Euler-Mascheroni constant and p 0 = γ / ( 1 − γ ) = 1.365 … $p_{0}=\gamma/(1-\gamma )=1.365\ldots$ .

In this paper, we present the best possible parameter $a\in(1/15, \infty)$ such that the functions $\psi^{\prime}(x+1)-\mathcal{L}_{x}(x, a)$ and $\psi^{\prime\prime }(x+1)-\mathcal{L}_{xx}(x, a)$ are strictly increasing or decreasing with respect to $x\in(0, \infty)$ , where $\mathcal{L}(x,a)=\frac {1}{90a^{2}+2}\log (x^{2}+x+\frac{3a+1}{3} )+ \frac {45a^{2}}{90a^{2}+2}\log (x^{2}+x+\frac{15a-1}{45a} )$ and $\psi(x)$ is the classical psi function. As applications, we get several new sharp bounds for the psi function and its derivatives.

In the paper, by virtue of two new tools, the authors present the monotonicity and convexity of certain combinations of the complete elliptic integrals of the first kind, and obtain new sharp bounds and inequalities for them.