Iranian Mathematical SocietyJournal of the Iranian Mathematical Society2717-16122120210301Sanov's theorem on Lie relators in groups of exponent $p$11611085610.30504/jims.2020.110856ENM.Vaughan-LeeChrist Church, University of Oxford,
Oxford, OX1 1DP, England.Journal Article20200515I 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 SocietyJournal of the Iranian Mathematical Society2717-16122120210301Generalized trapezoid type inequalities for functions with values in Banach spaces173813834110.30504/jims.2021.299742.1038ENS. S.DragomirVictoria University, Melbourne, AustraliJournal Article20210814Let 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 SocietyJournal of the Iranian Mathematical Society2717-16122120210301On a Hilbert-type integral inequality in the whole plane395114119910.30504/jims.2021.309063.1044ENM. Th.RassiasDepartment of Mathematics and Engineering Sciences,
Hellenic Military Academy,
16673 Vari Attikis, GreeceB.YangDepartment of Mathematics, Guangdong University of
Education, Guangzhou, Guangdong
510303, P. R. ChinaG. C.MeletiouUniversity of Ioannina, Ioannina, Greece.Journal Article20211005Using 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 SocietyJournal of the Iranian Mathematical Society2717-16122120210301The first eigenvalue of $left(p,qright)$-elliptic quasilinear system along the Ricci flow537014149010.30504/jims.2021.263742.1026ENS.AzamiDepartment of pure mathematics, Faculty of mathematical science, Imam Khomeini international university, Qazvin, IranM.Habibi Vosta KolaeiDepartment of pure mathematics, Faculty of science, Imam Khomeini international university, Qazvin, IranJournal Article20201225In 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.