On the one spectral relation for the analytic function of operator

In this work, some estimates for the difference number between operator norm and spectral radius of analytic functions of linear bounded Hilbert space operators via difference numbers of powers of corresponding Hilbert space operators have been obtained. Firstly, these evaluations for the polynomial functions of the linear bounded Hilbert space operator have been established. Using previous results, this question was later investigated for the exponential, sine, and cosine functions of a given operator. Finally, starting from obtained results, this subject for the analytic functions of the linear bounded Hilbert space operator has been generalized.