Iranian Mathematical Society Journal of the Iranian Mathematical Society 2717-1612 2 1 2021 03 01 Sanov's theorem on Lie relators in groups of exponent \$p\$ 1 16 110856 10.30504/jims.2020.110856 EN M. Vaughan-Lee Christ Church, University of Oxford, Oxford, OX1 1DP, England. Journal Article 2020 05 15 ‎I give a proof of Sanov's theorem that the Lie relators of weight at most‎ ‎\$2p-2\$ in groups of exponent \$p\$ are consequences of the identity \$px=0\$ and‎ ‎the \$(p-1)\$-Engel identity‎. ‎This implies that the order of the class \$2p-2\$‎ ‎quotient of the Burnside group \$B(m,p)\$ is the same as the order of the class‎ ‎\$2p-2\$ quotient of the free \$m\$ generator \$(p-1)\$-Engel Lie algebra over‎ ‎GF\$(p)\$‎. ‎To make the proof self-contained I have also included a derivation of‎ ‎Hausdorff's formulation of the Baker Campbell Hausdorff formula‎.
Iranian Mathematical Society Journal of the Iranian Mathematical Society 2717-1612 2 1 2021 03 01 Generalized trapezoid type inequalities for functions with values in Banach spaces 17 38 138341 10.30504/jims.2021.299742.1038 EN S. S. Dragomir Victoria University, Melbourne, Australi Journal Article 2021 08 14 Let E be a complex Banach space. In this paper we show among others that, if α:[a,b]→C is continuous and Y:[a,b]→E is strongly differentiable on the interval (a,b), then for all u∈[a,b],<br />‖(∫_{u}^{b}α(s)ds)Y(b)+(∫_{a}^{u}α(s)ds)Y(a)-∫_{a}^{b}α(t)Y(t)dt‖ ≤{┊max{∫_{u}^{b}|α(s)|ds,∫_{a}^{u}|α(s)|ds}∫_{a}^{b}‖Y′(t)‖dt, [∫_{u}^{b}(b-t)|α(t)|dt+∫_{a}^{u}(t-a)|α(t)|dt]sup_{t∈[a,b]}‖Y′(t)‖,  ≤(b-a)^{1/p}[(∫_{u}^{b}|α(s)|ds)^{p}+(∫_{a}^{u}|α(s)|ds)^{p}]^{1/p} <br />×(∫_{a}^{b}‖Y′(t)‖^{q}dt)^{1/q}<br /><br />for p, q>1 with (1/p)+(1/q)=1. Applications for operator monotone functions with examples for power and logarithmic functions are also given.
Iranian Mathematical Society Journal of the Iranian Mathematical Society 2717-1612 2 1 2021 03 01 On a Hilbert-type integral inequality in the whole plane 39 51 141199 10.30504/jims.2021.309063.1044 EN M. Th. Rassias Department of Mathematics and Engineering Sciences, Hellenic Military Academy, 16673 Vari Attikis, Greece B. Yang Department of Mathematics, Guangdong University of Education, Guangzhou, Guangdong 510303, P. R. China G. C. Meletiou University of Ioannina‎, ‎Ioannina‎, ‎Greece. Journal Article 2021 10 05 Using weight functions and techniques of real analysis, a new Hilbert-type integral inequality in the whole plane with nonhomogeneous kernel and a best possible constant factor is proved. Equivalent forms, several particular inequalities and operator expressions are considered.
Iranian Mathematical Society Journal of the Iranian Mathematical Society 2717-1612 2 1 2021 03 01 The first eigenvalue of \$left(p,qright)\$-elliptic quasilinear system along the Ricci flow 53 70 141490 10.30504/jims.2021.263742.1026 EN S. Azami Department of pure mathematics, Faculty of mathematical science, Imam Khomeini international university, Qazvin, Iran M. Habibi Vosta Kolaei Department of pure mathematics, Faculty of science, Imam Khomeini international university, Qazvin, Iran Journal Article 2020 12 25 In this paper we investigate the monotonicity of the first eigenvalue of \$left(p,qright)\$-elliptic quasilinear systems along the Ricci flow in both normalized and unnormalized conditions. In particular, we study the eigenvalue problem for this system in the case of Bianchi classes for \$3\$-homogeneous manifolds.