Let S be a smoothly bounded region in the complex plane. Let g ( z , t ) g(z,t) denote the Green’s function of S with pole at t. We show that \[ ∬ S | f ′ ( z ) | 2 d x d y ⩽ 1 2 ∫ ∂ S | f ′ ( z ) | 2 ( ∂ g ( z , t ) ∂ n z ) − 1 | d z | \iint _S {|f’(z){|^2}\,dx\,dy\, \leqslant \,\frac {1}{2}\int _{\partial S} {|f’(z){|^2}{{\left ( {\frac {{\partial g(z,t)}} {{\partial {n_z}}}} \right )}^{ - 1}}|dz|} } \] holds for any analytic function f ( z ) f(z) on S ∪ ∂ S S\, \cup \,\partial S . This curious inequality is obtained as a special case of a much more general result.