This paper introduces an adaptive numerical integration method specifically designed for functions of class C^6[a,b] with a sixth derivative of constant sign. The approach is based on a refined inequality involving the three-point Gauss quadrature ( 𝒢 ) and the four-point Lobatto quadrature ( ℒ ). We utilize a specific linear combination, Q = 3/4𝒢 + 1/4ℒ , and demonstrate that for functions with a sixth derivative of constant sign, the approximation error is effectively controlled by the difference between these two quadratures. We provide a rigorous justification for the stopping criterion of the adaptive algorithm. Numerical experiments, including the approximation of the integrals of a reciprocal function as well as the exponential function, show that the proposed method significantly outperforms existing adaptive techniques designed for lower-order convexity, requiring substantially fewer subintervals to achieve high precision (up to 10^-16 ).
Let S: [0, 1]^n →ℝ satisfy the identity anchoring condition S(e_i(t)) = t , where e_i(t) is the vertex with t in position i and 1 elsewhere. Suppose there exists ε≥ 0 such that S satisfies the four-point multiplicative separability condition approximately: |S(x) · S(y) - S(x ⊠ _i y) · S(y ⊠ _i x) |≤ε for every coordinate i and every pair (x, y) ∈ [0, 1]^n × [0, 1]^n , where x ⊠ _i y denotes the vector x with its i -th coordinate replaced by y_i , and 1 = (1, … , 1) denotes the all-ones vertex of the cube. We prove the uniform absolute bound valid on the entire closed cube, including boundary faces where ∏ _i x_i = 0 , and requiring no continuity, positivity, or small- ε hypothesis on S . The proof combines a one-coordinate reduction |S(x) - x_i · S(x ⊠ _i 1) |≤ε , obtained by a single application of the four-point identity to the pair (x, 1) , with a finite induction on the coordinate index. An explicit family attains a deviation of exactly ε for every n ≥ 2 , so the order O(ε ) is optimal and the sharp constant K_n satisfies 1 ≤ K_n ≤ n - 1 ; in particular K_2 = 1 . A classical Hyers–Ulam approach via logarithmic reduction yields a complementary relative bound on interior regions bounded away from the boundary (§4 bis). Section 7 situates the result in the Hyers–Ulam literature and states open problems.
Difference analogues of the logarithmic derivative lemma play an important part in investigating the value distribution of complex difference polynomials and determining the existence of meromorphic solutions of complex difference equations. In this paper, we establish several variants on the difference analogues of the logarithmic derivative lemma. Specifically, we obtain m( r,f(z+c_1)f(z+c_2)⋯ f(z+c_n)-a/f(z)^n-a) =S(r,f), where f(z) is a transcendental meromorphic function belonging to a specific function class and n is a positive integer. Subsequently, we demonstrate some applications of the variants of difference analogues of the logarithmic derivative lemma by establishing uniqueness theorems of value sharing and value distribution properties of complex difference polynomials.
In a recent paper, Matkowski conjectured that every weakly associative two-variable formal power series with non-symmetric linear part must be linear. In this note we disprove that conjecture. More precisely, we show that there exists a countable exceptional set of resonance parameters ℛ⊆𝕂 so that, for every parameter a∈𝕂∖ℛ and every prescribed quadratic coefficient c_2∈𝕂 , we construct a unique formal power series M(x,y)=y+g(x-y)∈𝕂[[x,y]] whose linear part is ax+(1-a)y and which satisfies the weak associativity equation M(M(x,y),x)=M(x,M(y,x)). If c_2 0 , then the resulting series is non-linear. We also analyze several resonance cases. We prove that the cases a=0 and a=1 are rigid, while the resonance value a=1/2 admits only symmetric formal solutions. Moreover, for every real parameter a∈ (0,1)∖(ℛ∪{12}), we obtain a non-linear convergent weakly associative power series with linear part ax+(1-a)y , which yields a non-linear real-analytic weakly associative strict mean on a sufficiently small interval. Thus Matkowski’s conjecture fails in the formal category, and the answer to its convergent counterpart is negative.
This paper introduces a new class of means named the more generalized Bajraktarević mean, which extends and unifies several classical means such as the generalized weighted quasi-arithmetic means and the generalized weighted Bajraktarević means. Based on this new definition, we further establish its set-valued extension and present several illustrative examples. Moreover, we derive sufficient conditions that ensure the Schur-concavity of the proposed more generalized Bajraktarević set-valued means.
