If `x and y` are positive real numbers and `m, n` are any positive integers, then prove that `(x^n y^m)/((1+x^(2n))(1+y^(2m))) lt 1/4`

If x,y are positive real numbers and m, n are positive integers, then prove that (x^(n) y^(m))/((1 + x^(2n))(1 + y^(2m))) le (1)/(4)

Let x, y be positive real numbers and m, n be positive integers, The maximum value of the expression (x^(m)y^(n))/((1+x^(2m))(1+y^(2n))) is

If n is a positive integer, prove that |I m(z^n)|lt=n|I m(z)||z|^(n-1.)

If m and n are positive quantites , prove that ((mn+1)/(m+1))^(m+1)ge n^(m)

For any positive real number a and for any n in N , the greatest value of (a^(n))/(1+a+a^(2)+…+a^(2n)) , is

If and n are positive integers, then lim_(x->0)((cosx)^(1/ m)-(cosx)^(1/ n))/(x^2) equal to :

If a ,\ m ,\ n are positive integers, then {root(m)root (n)a}^(m n) is equal to (a) a^(m n) (b) a (c) a^(m/n) (d) 1

For any positive real number x , prove that there exists an irrational number y such that 0 lt y lt x .

If x cos A + y sin A = m and x sin A - y cos A = n, then prove that : x^(2) + y^(2) = m^(2) + n^(2)

Prove that if xa n dy are odd positive integers, then x^2+y^2 is even but not divisible by 4.