I have found a theorem (Rouché's theorem) that uses the strict inequality and which might be of help here:
Suppose [imath] f [/imath] and [imath] g [/imath] are meromorphic in a neighborhood of [imath] \overline{B}(a;R) [/imath] with no zeros ( [imath]Z [/imath]) or poles ( [imath]P [/imath]) on the circle [imath]\gamma =\{z\in \mathbb{C}\,|\,|z-a|=R\} [/imath]. If [imath]Z_f,Z_g,P_f,P_g [/imath] are the numbers of zeros, resp. poles, of [imath] f [/imath] and [imath] g [/imath] inside [imath]\gamma [/imath] counted according to their multiplicities and if
[math] |f(z)+g(z)|<|f(z)|+|g(z)| [/math]on [imath] \gamma , [/imath] then
[math]
Z_f-P_f=Z_g-P_g\;.
[/math]Proof: From the hypothesis
[math]
\left|\dfrac{f(z)}{g(z)+1}\right|< \left|\dfrac{f(z)}{g(z)}\right|+1
[/math]on [imath]\gamma . [/imath] If [imath]\lambda :=f(z)/g(z) [/imath] and if [imath]\lambda [/imath] is a positive real number then this inequality becomes [imath] \lambda +1<\lambda +1, [/imath] a contradiction. Hence the meromorphic function [imath]f/g [/imath] maps [imath]\gamma [/imath] onto [imath]\Omega:=\mathbb{C}-[0,\infty ). [/imath] If [imath] L [/imath] is a branch of the logarithm on [imath]\Omega [/imath] then [imath]L(f(z)(g(z)) [/imath] is a well-defined primitive for [imath](f/g)\,'(f/g)^{-1} [/imath] in a neighborhood of [imath]\gamma . [/imath] Thus
[math]\begin{array}{lll}
0&= \dfrac{1}{2\pi i} \int_\gamma (f/g)\,'(f/g)^{-1} \\[12pt]
&= \dfrac{1}{2\pi i} \int_\gamma \left(\dfrac{f\,'}{f}-\dfrac{g\,'}{g}\right)\\[12pt]
&= (Z_f-P_f)-(Z_g-P_g).
\end{array}[/math]
