Sanov's theorem on Lie relators in groups of exponent $p$ M. Vaughan-Lee Christ Church, University of Oxford, Oxford, OX1 1DP, England. author text article 2021 eng ‎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. Journal of the Iranian Mathematical Society Iranian Mathematical Society 2717-1612 2 v. 1 no. 2021 1 16 http://jims.ims.ir/article_110856_87c80830a05958c113d30d5f05f1c835.pdf dx.doi.org/10.30504/jims.2020.110856 Generalized trapezoid type inequalities for functions with values in Banach spaces S. Dragomir Victoria University, Melbourne, Australi author text article 2021 eng 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],‖(∫_{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} ×(∫_{a}^{b}‖Y′(t)‖^{q}dt)^{1/q}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. Journal of the Iranian Mathematical Society Iranian Mathematical Society 2717-1612 2 v. 1 no. 2021 17 38 http://jims.ims.ir/article_138341_5a034622d668c9097177e956cd4e849e.pdf dx.doi.org/10.30504/jims.2021.299742.1038 On a Hilbert-type integral inequality in the whole plane M. Rassias Department of Mathematics and Engineering Sciences, Hellenic Military Academy, 16673 Vari Attikis, Greece author B. Yang Department of Mathematics, Guangdong University of Education, Guangzhou, Guangdong 510303, P. R. China author G. Meletiou University of Ioannina‎, ‎Ioannina‎, ‎Greece. author text article 2021 eng 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. Journal of the Iranian Mathematical Society Iranian Mathematical Society 2717-1612 2 v. 1 no. 2021 39 51 http://jims.ims.ir/article_141199_a726440e0929ea956c8fa65786aab4ca.pdf dx.doi.org/10.30504/jims.2021.309063.1044 The first eigenvalue of $\left(p,q\right)$-elliptic quasilinear system along the Ricci flow S. Azami Department of pure mathematics, Faculty of mathematical science, Imam Khomeini international university, Qazvin, Iran author M. Habibi Vosta Kolaei Department of pure mathematics, Faculty of science, Imam Khomeini international university, Qazvin, Iran author text article 2021 eng In this paper we investigate the monotonicity of the first eigenvalue of $\left(p,q\right)$-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. Journal of the Iranian Mathematical Society Iranian Mathematical Society 2717-1612 2 v. 1 no. 2021 53 70 http://jims.ims.ir/article_141490_c8292bfa33955943da919d35e5f22c90.pdf dx.doi.org/10.30504/jims.2021.263742.1026