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In this paper, we prove that the double inequality Mα(a,b) < NGQ(a,b) 0 with a ≠ b if and only if α ≤ 2log2/(5log2-2logπ) = 1:1785... and β ≥ 4/3,where NGQ(a,b) = [G(a,b)CQ2(a,b)=U(a,b)]/2 is the second Neuman mean, G(a,b) = √ab, Q(a,b) = √(a2+b2)/2 and U(a,b) = (a-b)/[√2tan-1((a - b)/ √2ab)] are the geometric, quadratic and Yang mean of a and b, respectively( [ABSTRACT FROM AUTHOR]

In this paper we have introduced the concept of geometric convergence of a sequence and determined the necessary and sufficient condition under which convergence follows from geometric convergence of a sequence in multiplicative sense. Corollaries allow this condition to be replaced by multiplicative analogues of Schmidt type slow oscillation condition or Landau type two-sided condition. [ABSTRACT FROM AUTHOR]