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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 ) hold for all x > 0 if and only if a ≥ 1 / 4 and b = 0 if a , b ∈ [ 0 , ∞ ) , where K ν ( x ) is the modified Bessel function of the second kind. As applications, we provide bounds for K n + 1 ( x ) / K n ( x ) with n ∈ N and present the necessary and sufficient condition such that the function x ↦ x + p e x K 0 ( x ) is strictly increasing (decreasing) on ( 0 , ∞ ) .

In this paper, by using the monotonicity rule for the ratio of two Laplace transforms, we prove that the function x ↦ 1 24 x ( ln Γ ( x + 1 / 2 ) − x ln x + x − ln 2 π ) + 1 − 120 7 x 2 is strictly increasing from ( 0 , ∞ ) onto ( 1 , 1860 / 343 ) . This not only yields some known and new inequalities for the gamma function, but also gives some completely monotonic functions related to the gamma function.

In the article, we present several sharp bounds for the modified Bessel function of the first kind I 0 ( t ) = ∑ n = 0 ∞ t 2 n 2 2 n ( n ! ) 2 and the Toader-Qi mean T Q ( a , b ) = 2 π ∫ 0 π / 2 a cos 2 θ b sin 2 θ d θ for all t > 0 and a , b > 0 with a ≠ b .

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 holds for all x ∈ ( 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 for all x ∈ ( 0 , 1 ) , and we find the value of x ∗ in the interval ( 0 , 1 ) such that Γ ( x + 1 ) > ( x 2 + 1 / γ ) / ( x + 1 / γ ) for x ∈ ( 0 , x ∗ ) and Γ ( x + 1 ) < ( x 2 + 1 / γ ) / ( x + 1 / γ ) for x ∈ ( x ∗ , 1 ) , where Γ ( x ) is the classical gamma function, γ = lim n → ∞ ( ∑ k = 1 n 1 / k − log n ) = 0.577 … is Euler-Mascheroni constant and p 0 = γ / ( 1 − γ ) = 1.365 …  .

In the paper, the authors introduce two notions, the normalized remainders, or say, the normalized tails, of the Maclaurin power series expansions of the sine and cosine functions, derive two integral representations of the normalized tails, prove the nonnegativity, positivity, decreasing property, and concavity of the normalized tails, compute several special values of the Young function, the Lommel function, and a generalized hypergeometric function, recover two inequalities for the tails of the Maclaurin power series expansions of the sine and cosine functions, propose three open problems about the nonnegativity, positivity, decreasing property, and concavity of a newly introduced function which is a generalization of the normalized tails of the Maclaurin power series expansions of the sine and cosine functions. These results are related to the Riemann–Liouville fractional integrals.

For $r\\in(0,1)$, let $\\mu \\left( r\\right) $ be the modulus of the plane Grötzsch ring $\\mathbb{B}^2\\setminus[0,r]$, where $\\mathbb{B}^2$ is the unit disk. In this paper, we prove that \\begin{equation*} \\mu \\left( r\\right) =\\ln \\frac{4}{r}-\\sum_{n=1}^{\\infty }\\frac{\\theta _{n}}{ 2n}r^{2n}, \\end{equation*} with $\\theta _{n}\\in \\left( 0,1\\right)$. Employing this series expansion, we obtain several absolutely monotonic and (logarithmically) completely monotonic functions involving $\\mu \\left( r\\right) $, which yields some new results and extend certain known ones. Moreover, we give an affirmative answer to the conjecture proposed by Alzer and Richards in H. Alzer and K. Richards, On the modulus of the Grötzsch ring, J. Math. Anal. Appl. 432(1): (2015), 134–141, DOI 10.1016/j.jmaa.2015.06.057. As applications, several new sharp bounds and functional inequalities for $\\mu \\left( r\\right) $ are established.

In the article, we provide a monotonicity rule for the function [ P ( x ) + A ( x ) ] / [ P ( x ) + B ( x ) ] , where P ( x ) is a positive differentiable and decreasing function defined on ( − R , R ) ( R > 0 ), and A ( x ) = ∑ n = n 0 ∞ a n x n and B ( x ) = ∑ n = n 0 ∞ b n x n are two real power series converging on ( − R , R ) such that the sequence { a n / b n } n = n 0 ∞ is increasing (decreasing) with a n 0 / b n 0 ≥ ( ≤ ) 1 and b n > 0 for all n ≥ n 0 . As applications, we present new bounds for the complete elliptic integral E ( r ) = ∫ 0 π / 2 1 − r 2 sin 2 t d t ( 0 < r < 1 ) of the second kind.

In the article, we prove that the function r ↦ E ( r ) / S 9 / 2 − p , p ( 1 , r ′ ) is strictly increasing on ( 0 , 1 ) for p ≤ 7 / 4 and strictly decreasing on ( 0 , 1 ) for p ∈ [ 2 , 9 / 4 ] , where r ′ = 1 − r 2 , E ( r ) = ∫ 0 π / 2 1 − r 2 sin 2 ( t ) d t is the complete elliptic integral of the second kind, and S p , q ( a , b ) = [ q ( a p − b p ) / ( p ( a q − b q ) ) ] 1 / ( p − q ) is the Stolarsky mean of a and b . As applications, we present several new bounds for E ( r ) , the Toader mean T ( a , b ) = ( 2 / π ) ∫ 0 π / 2 a 2 cos 2 t + b 2 sin 2 t d t , and the Toader-Qi mean TQ ( a , b ) = ( 2 / π ) ∫ 0 π / 2 a cos 2 θ b sin 2 θ d θ .

Let K ( r ) be the complete elliptic integral of the first kind. We present an accurate rational lower approximation for K ( r ) . More precisely, we establish the inequality 2 π K ( r ) > 5 ( r ′ ) 2 + 126 r ′ + 61 61 ( r ′ ) 2 + 110 r ′ + 21 for r ∈ ( 0 , 1 ) , where r ′ = 1 − r 2 . The lower bound is sharp.