An extension of the Euclidean Berezin number

The Berezin transform A of an operator A, acting on the reproducing kernel Hilbert space H = H(O) over some (non-empty) set O, is defined by A (& lambda;) = & LeftAngleBracket;Ak<SIC>& lambda;,k<SIC>& lambda;& RightAngleBracket; (& lambda; & ISIN; O), where k<SIC>& lambda; = k & lambda; & PAR;k & lambda;& PAR; is the normalized reproducing kernel of H. The Berezin number of an operator A is defined by ber(A) = sup & lambda;& ISIN;O we obtain some possible relations and inequalities. It is shown, among other inequalities, that if Ai & ISIN; L(H(O)) (i = 1, ...,n), then divide A (& lambda;) divide = sup divide & LeftAngleBracket;Ak<SIC>& lambda;, k<SIC>& lambda; & RightAngleBracket; divide . In this paper, by using the definition of 1-generalized Euclidean Berezin number, & lambda;& ISIN;O n nber1(A1,..., An) & LE; 1-1 ( n-ary sumation 1( ber(Ai))) & LE; n-ary sumation ber(Ai), i=1 i=1in which 1 & COLRATIO; [0, & INFIN;) & RARR; [0, & INFIN;) is a continuous increasing convex function such that 1(0) = 0.