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In the article, we establish a sharp double inequality involving the ratio of generalized complete elliptic integrals of the first kind, which is the improvement and generalization of some previously known results.

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 this paper, we introduce the notion of completely mixed monotonicity of a function of several variables, very few of which have appeared. We give a necessary and sufficient condition for a function constructed by ratios of gamma functions to be completely mixed monotonic. From this, some new inequalities for gamma, psi, and polygamma functions are derived.

We provide several sharp upper and lower bounds for the generalized Euler–Mascheroni constant. As consequences, some previous bounds for the Euler–Mascheroni constant are improved.

In this paper, we present the best possible parameter a ∈ ( 1 / 15 , ∞ ) such that the functions ψ ′ ( x + 1 ) − L x ( x , a ) and ψ ″ ( x + 1 ) − L x x ( x , a ) are strictly increasing or decreasing with respect to x ∈ ( 0 , ∞ ) , where L ( x , a ) = 1 90 a 2 + 2 log ( x 2 + x + 3 a + 1 3 ) + 45 a 2 90 a 2 + 2 log ( x 2 + x + 15 a − 1 45 a ) and ψ ( x ) is the classical psi function. As applications, we get several new sharp bounds for the psi function and its derivatives.

We systematically exploit a new generalized hypergeometric identity to obtain new hypergeometric summation formulas. As a consistency test, alternative proofs for some special cases are also provided. As a byproduct, new summation formulas with finite sums involving the psi function and a recursive formula for Bateman’s G function are derived. Finally, all the results have been numerically checked with MATHEMATICA.

Let Γ(x) denote the classical Euler gamma function. We set ψn(x)=(−1)n−1ψ(n)(x) (n∈N), where ψ(n)(x) denotes the nth derivative of the psi function ψ(x)=Γ′(x)/Γ(x). For λ, α, β∈R and m,n∈N, we establish necessary and sufficient conditions for the functions L(x;λ,α,β)=ψm+n(x)−λψm(x+α)ψn(x+β) and −L(x;λ,α,β) to be completely monotonic on (−min(α,β,0),∞).As a result, we generalize and refine some inequalities involving the polygamma functions and also give some inequalities in terms of the ratio of gamma functions.

Machine learning plays a pivotal role in addressing real-world challenges across domains such as cybersecurity, where AI-driven methods, especially in Software-Defined Networking, enhance traffic monitoring and anomaly detection. Contemporary networks often employ models like Random Forests, Neural Networks, and Support Vector Machines to identify threats early and reinforce security. Ensemble learning further improves predictive accuracy and stability, yet many frameworks falter when confronted with noisy or contaminated data. In this study, we propose a robust ensemble framework for Extreme Learning Machines (ELMs) that integrates a family of redescending ψ-activation functions grounded in M-estimation theory. Each ψ-function yields a distinct base classifier, initialized with random weights, and the optimal hidden-node count is selected via grid search minimizing the Brier score. Rather than traditional voting, ensemble outputs are combined through a least-squares optimization, allowing precise parameter estimation and enhanced stability. We validate our method on five benchmark datasets, SatImage, Email-Spamdexing, Breast Cancer, Musk, and Iris, demonstrating consistently superior accuracy and reduced variance compared to existing ELM ensembles. Rigorous statistical testing (Kruskal–Wallis with Dunn’s post-hoc comparisons) confirms these gains. Our results show that embedding robust M-estimator–based activations within a controlled ensemble yields marked improvements in generalization, predictive precision, and resilience to data irregularities, offering a significant advancement in the design of efficient neural classifiers.

For a ∈ (0, 1/2] and r ∈ (0, 1), let 𝒦ₐ(r) (𝒦 (r)) denote the generalized elliptic integral (complete elliptic integral, respectively) of the first kind. In this article, we mainly present a sufficient and necessary condition under which the function a 7 ↦ [𝒦 (r) − 𝒦ₐ(r)]/(1 − 2a)λ(λ ∈ ℝ) is monotone on (0, 1/2) for each fixed r ∈ (0, 1). The obtained result leads to the conclusion that inequality K ( r ) − ( 1 − 2 a ) α [ K ( r ) − π 2 ] ≤ K a ( r ) ≤ K ( r ) − ( 1 − 2 a ) β [ K ( r ) − π 2 ] holds for all a ∈ (0, 1/2] and r ∈ (0, 1) with the best possible constants α = π/2 and β = 2.

In this paper, two new approximation formulas with monotonic remainders for the gamma function have been presented. Also, we present some numerical comparisons between our new approximation formulas and some known ones, which demonstrate the superiority of our results.