Zhelnov O. Whitney inequality and its generalization

The objects of research are Whitney inequality, interpolation Whitney inequality, generalized Holder spaces. The goal of the research is to obtain estimates of Whitney constants and interpolatory Whitney constants for small k; to prove the existence of the linear bounded operator for extension on the real line of the spaces of functions which are together with their derivatives are restrictions of functions from generalized Holder spaces. The methods elaborated by H. Whitney, V.K. Dziadyk, E. A. Storozhenko, B. Sendov, Yu. V. Kryakin, I.A. Shevchuk where used and improved. Sendov conjecture that Whitney constants are bounded by 1 is confirmed for k=5,6,7,8. Sendov conjecture that Whitney interpolatory constants are bounded by 2 is confirmed for k=5,6,7,8. The estimates of Whitney constants for k=3,4 are improved. The linear bounded operator for extension on the real line of the spaces of functions which are together with their derivatives are restrictions of functions from generalized Holder spaces isconstructed. The field of application is the approximation of functions by splines and polynomials.