On an extension of Nunokawa’s lemma

Jack’s Lemma says that if f(z) is regular in the disc $$|z|\le r$$ | z | ≤ r , $$f(0)=0$$ f ( 0 ) = 0 , and |f(z)| assumes its maximum at $$z_0$$ z 0 on the circle $$|z|=r$$ | z | = r , then $$z_0f'(z)_0/f(z_0)\ge 1$$ z 0 f ′ ( z ) 0 / f ( z 0 ) ≥ 1 . This Lemma was generalized in several directions. In this paper we consider an improvement of some first author’s results of this type.

Hadamard’s variational formula in terms of stress and strain tensors

Starting from a Lagrangian action functional for two scalar fields we construct, by variational methods, the Laplacian Green function for a bounded domain and an appropriate stress tensor. By a further variation, imposed by a given vector field, we arrive at an interior version of the Hadamard variational formula, previously considered by P. Garabedian. It gives the variation of the Green function in terms of a pairing between the stress tensor and a strain tensor in the interior of the domain, this contrasting the classical Hadamard formula which is expressed as a pure boundary variation.