**MSC:**- 47H09 Nonexpansive mappings, and their generalizations
- 47H19 Inequalities involving nonlinear operators, See Also {49J27, 49J40, 49K27}

Then it is shown that there exists an increasing continuous function $h :

\Bbb R^+ \rightarrow \Bbb R^+$ depending on the diameter of $C$ so that for

any nonexpansive mapping $T:C \rightarrow X$ and any convex combination of

arbitrarily many elements $x_i$ in $C$ the inequality $h (\|\sum \lambda_i

x_i - T (\sum \lambda_i x_i)\|) \le \sum \lambda_i \| x_i - Tx_i \|$ holds.

This

inequality has several consequences in the theory of nonexpansive

mappings.