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A functional inequality related to Domar’s uniform boundedness theorem 与多玛一致有界性定理有关的一个泛函不等式
IF 0.9 3区 数学
Aequationes Mathematicae Pub Date : 2026-10-15 DOI: 10.1007/s00010-026-01334-1
Thomas Ransford
{"title":"A functional inequality related to Domar’s uniform boundedness theorem","authors":"Thomas Ransford","doi":"10.1007/s00010-026-01334-1","DOIUrl":"10.1007/s00010-026-01334-1","url":null,"abstract":"<div><p>We study the functional inequality </p><div><div><span>$$ f(r+s)le g(r)+alpha f(s) quad (r,s&gt;0). $$</span></div></div><p>Here <span>(g:(0,infty )rightarrow [0,infty ))</span> is a given decreasing function, <span>(alpha )</span> is a constant such that <span>(0&lt;alpha &lt;1)</span>, and the problem is to determine whether the family of decreasing functions <span>(f:(0,infty )rightarrow [0,infty ))</span> that satisfy this inequality is bounded above by some finite function on <span>((0,infty ))</span> and, if so, to find bounds for this function. We present a solution to this problem, and use it to give a new proof of a theorem of Domar on the uniform boundedness of certain families of subharmonic functions, while also obtaining explicit bounds.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"100 6","pages":""},"PeriodicalIF":0.9,"publicationDate":"2026-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149189765","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Adaptive integration of functions maintaining a constant sign of the sixth derivative 函数的自适应积分保持六阶导数的常数
IF 0.9 3区 数学
Aequationes Mathematicae Pub Date : 2026-10-09 DOI: 10.1007/s00010-026-01333-2
Szymon Wa̧sowicz
{"title":"Adaptive integration of functions maintaining a constant sign of the sixth derivative","authors":"Szymon Wa̧sowicz","doi":"10.1007/s00010-026-01333-2","DOIUrl":"10.1007/s00010-026-01333-2","url":null,"abstract":"<div><p>This paper introduces an adaptive numerical integration method specifically designed for functions of class <span>(C^6[a,b])</span> with a sixth derivative of constant sign. The approach is based on a refined inequality involving the three-point Gauss quadrature (<span>(mathcal {G})</span>) and the four-point Lobatto quadrature (<span>(mathcal {L})</span>). We utilize a specific linear combination, <span>(Q = frac{3}{4}mathcal {G} + frac{1}{4}mathcal {L})</span>, 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 <span>(10^{-16})</span>).</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"100 6","pages":""},"PeriodicalIF":0.9,"publicationDate":"2026-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00010-026-01333-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149132765","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Hyers–Ulam stability of the multiplicative separability condition on ([0, 1]^n) 上的乘法可分性条件的Hyers-Ulam稳定性 ([0, 1]^n)
IF 0.9 3区 数学
Aequationes Mathematicae Pub Date : 2026-10-08 DOI: 10.1007/s00010-026-01322-5
David M. Dror, Gérard Duru
{"title":"Hyers–Ulam stability of the multiplicative separability condition on ([0, 1]^n)","authors":"David M. Dror,&nbsp;Gérard Duru","doi":"10.1007/s00010-026-01322-5","DOIUrl":"10.1007/s00010-026-01322-5","url":null,"abstract":"<div><p>Let <span>(S: [0, 1]^n rightarrow mathbb {R})</span> satisfy the identity anchoring condition <span>(S(e_i(t)) = t)</span>, where <span>(e_i(t))</span> is the vertex with <span>(t)</span> in position <span>(i)</span> and <span>(1)</span> elsewhere. Suppose there exists <span>(varepsilon ge 0)</span> such that <span>(S)</span> satisfies the four-point multiplicative separability condition approximately: </p><div><div><span>$$begin{aligned} |S(x) cdot S(y) - S(x boxtimes _i y) cdot S(y boxtimes _i x) |le varepsilon end{aligned}$$</span></div></div><p>for every coordinate <span>(i)</span> and every pair <span>((x, y) in [0, 1]^n times [0, 1]^n)</span>, where <span>(x boxtimes _i y)</span> denotes the vector <span>(x)</span> with its <span>(i)</span>-th coordinate replaced by <span>(y_i)</span>, and <span>(textbf{1} = (1, ldots , 1))</span> denotes the all-ones vertex of the cube. We prove the uniform absolute bound </p><div><div><img></div></div><p>valid on the entire closed cube, including boundary faces where <span>(prod _i x_i = 0)</span>, and requiring no continuity, positivity, or small-<span>(varepsilon )</span> hypothesis on <span>(S)</span>. The proof combines a one-coordinate reduction <span>(|S(x) - x_i cdot S(x boxtimes _i textbf{1}) |le varepsilon )</span>, obtained by a single application of the four-point identity to the pair <span>((x, textbf{1}))</span>, with a finite induction on the coordinate index. An explicit family attains a deviation of exactly <span>(varepsilon )</span> for every <span>(n ge 2)</span>, so the order <span>(O(varepsilon ))</span> is optimal and the sharp constant <span>(K_n)</span> satisfies <span>(1 le K_n le n - 1)</span>; in particular <span>(K_2 = 1)</span>. 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.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"100 6","pages":""},"PeriodicalIF":0.9,"publicationDate":"2026-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149132521","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Difference analogues of the logarithmic derivative lemma and their applications in value distribution 对数导数引理的差分类似物及其在值分布中的应用
IF 0.9 3区 数学
Aequationes Mathematicae Pub Date : 2026-10-06 DOI: 10.1007/s00010-026-01331-4
Kai Liu, Yuxin Fan
{"title":"Difference analogues of the logarithmic derivative lemma and their applications in value distribution","authors":"Kai Liu,&nbsp;Yuxin Fan","doi":"10.1007/s00010-026-01331-4","DOIUrl":"10.1007/s00010-026-01331-4","url":null,"abstract":"<div><p>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 </p><div><div><span>$$mleft( r,frac{f(z+c_{1})f(z+c_{2})cdots f(z+c_{n})-a}{f(z)^{n}-a}right) =S(r,f),$$</span></div></div><p>where <i>f</i>(<i>z</i>) is a transcendental meromorphic function belonging to a specific function class and <i>n</i> 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.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"100 6","pages":""},"PeriodicalIF":0.9,"publicationDate":"2026-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149098467","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Monotonic solutions of a composite functional equation 复合泛函方程的单调解
IF 0.9 3区 数学
Aequationes Mathematicae Pub Date : 2026-10-05 DOI: 10.1007/s00010-026-01330-5
Soichi Ikeda
{"title":"Monotonic solutions of a composite functional equation","authors":"Soichi Ikeda","doi":"10.1007/s00010-026-01330-5","DOIUrl":"10.1007/s00010-026-01330-5","url":null,"abstract":"<div><p>Chao first considered the composite functional equation </p><div><div><span>$$begin{aligned} f(x + 2f(y)) = f(x) + y + f(y) quad (x, y in mathbb {R}), end{aligned}$$</span></div></div><p>where <span>(f: mathbb {R} rightarrow mathbb {R})</span>. This equation was studied by several mathematicians. In this paper, we study the monotonic solutions of the composite functional equation </p><div><div><span>$$begin{aligned} f(xyf(y)) = f(x) y f(y) quad (x, y in mathbb {R}), end{aligned}$$</span></div></div><p>where <span>(f: mathbb {R} rightarrow mathbb {R})</span>. This equation is similar to Chao’s equation.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"100 6","pages":""},"PeriodicalIF":0.9,"publicationDate":"2026-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00010-026-01330-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149098281","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Hyperrectangular and hyperrhombic functional equations 超矩形和超菱形泛函方程
IF 0.9 3区 数学
Aequationes Mathematicae Pub Date : 2026-10-03 DOI: 10.1007/s00010-026-01329-y
Bruce Ebanks
{"title":"Hyperrectangular and hyperrhombic functional equations","authors":"Bruce Ebanks","doi":"10.1007/s00010-026-01329-y","DOIUrl":"10.1007/s00010-026-01329-y","url":null,"abstract":"<div><p>The classical rectangular and rhombic functional equations for functions mapping <span>({mathbb {R}}times {mathbb {R}})</span> into <span>({mathbb {R}})</span> were solved in an article (Aequat. Math. <b>1</b>, 37-53, 1968) by Aczél, H. Haruki, McKiernan and Sakovič about 60 years ago. The authors showed that the two equations are equivalent and their solutions of are the sum of a bi-additive function, additive functions of each variable, and a constant. A recent article (Aequat. Math. (2026) 100:69) by Lin, Yang and Zhao extended the results on the rectangular equation to hyperrectangular equations on groups. In the present article we generalize those results by solving the hyperrectangular and hyperrhombic functional equations on monoids with an involution.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"100 6","pages":""},"PeriodicalIF":0.9,"publicationDate":"2026-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149098103","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Estimates of the gap between extreme solutions of nonlinear difference equations 非线性差分方程极值解间隙的估计
IF 0.9 3区 数学
Aequationes Mathematicae Pub Date : 2026-09-30 DOI: 10.1007/s00010-026-01321-6
Gerd Herzog, Peer Kunstmann
{"title":"Estimates of the gap between extreme solutions of nonlinear difference equations","authors":"Gerd Herzog,&nbsp;Peer Kunstmann","doi":"10.1007/s00010-026-01321-6","DOIUrl":"10.1007/s00010-026-01321-6","url":null,"abstract":"<div><p>We study bounded solutions of nonlinear difference equations on arbitrary index sets within a unified fixed-point framework. Working in the Banach lattice <span>(l^infty (X,mathbb {R}))</span>, we consider equations of the form </p><div><div><span>$$ M(u) = lambda u + D_f(u) + w, $$</span></div></div><p>where <i>M</i> is an increasing linear operator, <span>(D_f)</span> is the diagonal operator induced by a scalar super-linear nonlinearity <span>(fin C^1(mathbb {R},mathbb {R}))</span>, and <span>(win l^infty (X,mathbb {R}))</span>. Our abstract results apply to a broad class of discrete models, including discrete Laplace-type operators on <span>(mathbb {R}^n)</span>, Banach limits and difference equations with spatial structure.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"100 5","pages":""},"PeriodicalIF":0.9,"publicationDate":"2026-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00010-026-01321-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149133224","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The arithmetic–geometric mean and Bohr-type inequalities for sectorial matrices 扇形矩阵的算术几何平均和玻尔型不等式
IF 0.9 3区 数学
Aequationes Mathematicae Pub Date : 2026-09-29 DOI: 10.1007/s00010-026-01324-3
Mohammad El-Haddad, Omar Hirzallah, Fuad Kittaneh
{"title":"The arithmetic–geometric mean and Bohr-type inequalities for sectorial matrices","authors":"Mohammad El-Haddad,&nbsp;Omar Hirzallah,&nbsp;Fuad Kittaneh","doi":"10.1007/s00010-026-01324-3","DOIUrl":"10.1007/s00010-026-01324-3","url":null,"abstract":"<div><p>In this paper, we establish arithmetic–geometric mean inequalities for products of sectorial matrices under arbitrary unitarily invariant norms. For <span>(alpha )</span>-sectorial operators <i>A</i>,<i>B</i> and a positive semidefinite matrix <i>X</i>, we bound the norm of <i>AXB</i> by an expression involving <span>((operatorname {Re}A)^{2})</span> and <span>((operatorname {Re}B)^{2})</span> with the multiplicative factor <span>(sec ^{2}alpha )</span>. We develop trace versions, extend the results to positive linear maps, and establish matrix Bohr inequalities for sectorial matrices. These results encompass the classical case of positive semidefinite matrices when <span>(alpha =0)</span>. We provide explicit equality examples demonstrating that our estimates are sharp.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"100 5","pages":""},"PeriodicalIF":0.9,"publicationDate":"2026-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149133289","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Polynomial maps on the monoid of words 多项式映射在词的单形上
IF 0.9 3区 数学
Aequationes Mathematicae Pub Date : 2026-09-29 DOI: 10.1007/s00010-026-01327-0
Jose María Almira
{"title":"Polynomial maps on the monoid of words","authors":"Jose María Almira","doi":"10.1007/s00010-026-01327-0","DOIUrl":"10.1007/s00010-026-01327-0","url":null,"abstract":"<div><p>We briefly visit the theory of polynomial and semipolynomial maps defined on an arbitrary monoid, with range a commutative group. Then we characterize the space <span>(mathcal {P}(S,mathbb {C}))</span> of polynomial maps <span>(f:Srightarrow mathbb {C})</span>, where <span>(S=mathcal {A}^*)</span> is the monoid of words based on an arbitrary alphabet <span>(mathcal {A})</span> under concatenation, and we prove that if there exists a monoid <i>S</i> such that <span>(mathcal{S}mathcal{P}(S,mathbb {C})ne mathcal {P}(S,mathbb {C}))</span>, then also <span>(mathcal{S}mathcal{P}(mathcal {A}^*,mathbb {C})ne mathcal {P}(mathcal {A}^*,mathbb {C}))</span> for a certain alphabet <span>(mathcal {A})</span>. We propose as an open problem to prove or disprove that <span>(mathcal{S}mathcal{P}(mathcal {A}^*,mathbb {C})=mathcal {P}(mathcal {A}^*,mathbb {C}))</span> for arbitrary alphabets <span>(mathcal {A})</span>. Finally, we apply our algebraic framework to combinatorial objects by proving that permutation pattern functions are polynomial maps. Our results are motivated by previous work of Shulman [3, 4].</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"100 5","pages":""},"PeriodicalIF":0.9,"publicationDate":"2026-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149133290","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Desargues’s and Pappus’s hexagon theorems on translation-type surfaces in Thurston geometries 瑟斯顿几何中平移型曲面上的des辩称和Pappus的六边形定理
IF 0.9 3区 数学
Aequationes Mathematicae Pub Date : 2026-09-26 DOI: 10.1007/s00010-026-01325-2
Jenő Szirmai
{"title":"Desargues’s and Pappus’s hexagon theorems on translation-type surfaces in Thurston geometries","authors":"Jenő Szirmai","doi":"10.1007/s00010-026-01325-2","DOIUrl":"10.1007/s00010-026-01325-2","url":null,"abstract":"<div><p>In [35] we generalized the famous Menelaus’ and Ceva’s theorems for translation triangles in each non-constant curvature Thurston geometry. In this paper based on the described method and results, we prove that the classical Desargues’s and Pappus’s hexagon theorems are true not only in classical geometries with constant curvature, but also in Thurston geometries with non-constant curvature on the translation surfaces. In our work we use the unified projective models of Thurston geometries.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"100 5","pages":""},"PeriodicalIF":0.9,"publicationDate":"2026-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00010-026-01325-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149133084","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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