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In this paper, we introduce some monotonicity rules for the ratio of integrals. Furthermore, we demonstrate that the function \\(-T_ , , (s)\\) is completely monotonic in \\(s\\) and absolutely monotonic in \\( \\) if and only if \\( 1\\) , where \\(T_ , , (s)=K_ ^2(s)- K_ - (s)K_ + (s)\\) defined on \\(s>0\\) and \\(K_ (s)\\) is the modified Bessel function of the second kind of order \\( \\) . Finally, we determine the necessary and sufficient conditions for the functions \\(s T_ , ,1(s)/T_ , ,1(s)\\) , \\(s (T_ , ,1(s) + T_ , ,1(s))/(2T_( + )/2, ,1(s))\\) , and \\(s d^n_1d ^n_1 T_ , ,1(s)/d^n_2d ^n_2 T_ , ,1(s)\\) to be monotonic in \\(sın (0,ınfty )\\) by employing the monotonicity rules.

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.

The study comprehensively evaluated the prognostic roles of the platelet-to-lymphocyte ratio (PLR), neutrophil-to-lymphocyte ratio (NLR), monocyte-to-lymphocyte ratio (MLR), basophil-to-lymphocyte ratio (BLR), and eosinophil-to-lymphocyte ratio (ELR) in patients with acute exacerbation of chronic obstructive pulmonary disease (AECOPD). Six hundred and nineteen patients with AECOPD and 300 healthy volunteers were retrospectively included into the study. The clinical characteristics of the patients with AECOPD and the complete blood counts (CBCs) of the healthy volunteers were collected. The associations of PLR, NLR, MLR, BLR, and ELR with airflow limitation, hospital length of stay (LOS), C-reactive protein (CRP), and in-hospital mortality in patients with AECOPD were analyzed. Compared with the healthy volunteers, PLR, NLR, MLR, BLR, and ELR were all elevated in COPD patients under stable condition. PLR, NLR, MLR, and BLR were further elevated while ELR was lowered during exacerbation. In the patients with AECOPD, PLR, NLR, and MLR were positively correlated with hospital LOS as well as CRP. In contrast, ELR was negatively correlated with hospital LOS as well as CRP. Elevated PLR, NLR, and MLR were all associated with more severe airflow limitation in AECOPD. Elevated PLR, NLR, and MLR were all associated with increased in-hospital mortality while elevated ELR was associated with decreased in-hospital mortality. Binary logistic regression analysis showed that smoking history, FEV1% predicted, pneumonia, pulmonary heart disease (PHD), uric acid (UA), albumin, and MLR were significant independent predictors ofin-hospital mortality. These predictors along with ELR were used to construct a nomogram for predicting in-hospital mortality in AECOPD. The nomogram had a C-index of 0.850 (95% CI: 0.799-0.901), and the calibration curve, decision curve analysis (DCA), and clinical impact curve (CIC) further demonstrated its good predictive value and clinical applicability. In summary, PLR, NLR, MLR, and ELR served as useful biomarkers in patients with AECOPD.

In this paper, we first introduce the definition of triple Diamond-Alpha integral for functions of three variables. Therefore, we present the Hölder and reverse Hölder inequalities for the triple Diamond-Alpha integral on time scales, and then we obtain some new generalizations of the Hölder and reverse Hölder inequalities for the triple Diamond-Alpha integral. Moreover, using the obtained results, we give a new generalization of the Minkowski inequality for the triple Diamond-Alpha integral on time scales.

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 this paper, we present some new Cauchy–Schwarz inequalities for N-tuple diamond-alpha integral on time scales. The obtained results improve and generalize some Cauchy–Schwarz type inequalities given by many authors.

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.

In this paper, we consider the oscillation behavior of the following second-order nonlinear dynamic equation. λ(s)Ψ1φΔ(s)y(φ(s))ΔΔ+η(s)Φ(y(τ(s)))=0,s∈[s0,∞)T. By employing generalized Riccati transformation and inequality scaling technique, we establish some oscillation criteria.

In this paper we develop Windschitl’s approximation formula for the gamma function by giving two asymptotic expansions using a little known power series. In particular, for ∈ ℕ with ≥ 4, we have with for all > 0, where has a closed-form expression, is the Bernoulli number. Moreover, we present some approximation formulas for the gamma function related to Windschitl’s approximation, which have higher accuracy.

Suppose w,v>0 , w≠v and Au(w,v) is the u -order power mean (PM) of w and v . In this paper, we completely describe the convexity of u↦Au(w,v) on R and with u(s)=(ln2)/ln(1/s) on (0,∞). These yield some new inequalities for PMs, and give an answer to an open problem.