The results of this paper are part of the contents of the author's Ph. D. thesis, done under the direction of Professor Leo Sario, to whom I wish toexpress my most sincere gratitude. ON EXTREMAL PROPERTIES FOR ANNULAR MAPPINGS 1491 COROLLARY 1. p On maximizes the functional 2πc{p)-\ pdp* among all p e H n. COROLLARY 2. p ln minimizes the functional 2πc(p) + \_ pdp* among all p e H n. COROLLARY 3.-(p On + p ln) minimizes the functional 2 Γβ 7t among all pe H n. COROLLARY 4. p on-p ln maximizes the functional Aπc{p)-D Wn (p) among all p e H n (0). Each extremizing function is unique.

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We propose an efficient eigenmode solver for electromagnetic cavities based on the Jacobi-Davidson (JD) method and hierarchical finite elements (FE) of high order. The JD algorithm enables us to exploit the multilevel structure of the FE space at various stages of the solution process. As a result, the computational complexity of the suggested method is very low, which is particularly useful for large-scale projects.