On the Zeros and Poles of a Meromorphic Function

Theorem 1 : If f(z) be a meromorphic function of order p, 0<p<1 , with all its zeros and poles real and negative and if Δ (r) be the excess of the zeros of f(z) over its poles, where we suppose that it is a non-decreasing function of r, for¦z¦ < r and if, for x → ∞ ,