Abstract.An elementary functional analytic argument is given showing how inequalities of the form \\f(k)\f < A^ll/H""*!!/00!!*, where/is a real, n-times differentiable function and ||-|| denotes the sup norm on (0, oo) (or (-00,00)), yield corresponding inequalities, \Akx\" < ^¿M"-\A"x\ , for generators of linear contraction semigroups (or groups) on arbitrary Banach spaces with norm ||. Since Landau, Kolmogorov, Schoenberg and Cavaretta have established the function inequalities with the best possible constants, this argument gives the generator inequalities with the best possible constants for general Banach spaces extending work of Kallman and Rota, Hille and others. Questions concerning the best possible constants for specific Banach spaces remain open.

Abstract.An elementary functional analytic argument is given showing how inequalities of the form \\f(k)\f < A^ll/H""*!!/00!!*, where/is a real, n-times differentiable function and ||-|| denotes the sup norm on (0, oo) (or (-00,00)), yield corresponding inequalities, \Akx\" < ^¿M"-\A"x\ , for generators of linear contraction semigroups (or groups) on arbitrary Banach spaces with norm ||. Since Landau, Kolmogorov, Schoenberg and Cavaretta have established the function inequalities with the best possible constants, this argument gives the generator inequalities with the best possible constants for general Banach spaces extending work of Kallman and Rota, Hille and others. Questions concerning the best possible constants for specific Banach spaces remain open.

Key words and phrases. Semigroup of operators, holomorphic semigroup, analytic extension of functions, finite differences, quasianalytic classes of functions.

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