Оценка нормы производной комплекснозначной функции с выпуклой областью изменения производной второго порядка

Many results related to the so-called comparison theorems and inequalities for derivatives in different classes of differentiable functions have been obtained in the theory of approximation of functions. In what follows we consider the class of differentiable functions with an absolutely continuous derivative on any straight-line segment and essentially restricted by a derivative of higher order. Our work [1] presented the evaluation of the actual performance of differentiable functions with asymmetrical restrictions on the second derivative. In paper [2] we provided the results extended to the class of complex-valued differentiable functions with asymmetric restrictions on the second derivative. We considered a case when the domain of variation of the second-order derivative was an ellipse with one of the focuses at the origin of coordinates. It is worth noting that the problem of evaluating the performance of real or complex-valued functions is related to the problem of estimating the norms of derivatives of such functions. It turned out that in this case the norm of the derivative restricted in the norm of complex-valued functions can be evaluated using Bernoulli splines applied in [5], or Euler splines [3]. Here we have received a bilateral evaluation of the derivative norm of a complex-valued differentiable function with asymmetric restrictions on the second-order derivative, namely, we have considered a case when the domain of variation of second-order derivative is a convex set of a complex plane. If we fit a certain ellipse with one of the focuses at the origin of coordinates in this set and describe another ellipse around this set, it is possible to obtain a two-sided evaluation of the norm of the restricted complex-valued function derivative. This raises a problem of finding ellipses that best encompass the boundary of the given convex set. To get the best inscribed ellipse we can use the maximization of the major semiaxis and the minimization of the distance from the origin of coordinates to the focus as a criterion. To get the best circumscribed ellipse we can use the minimization of the major semiaxis and the maximization of the distance from the origin of coordinates to the focus as a criterion. This problem solved, we will be able to minimize the difference between the upper and lower estimate of the derivative norm. Here we have obtained a bilateral evaluation of the derivative norm of the restricted complex-valued function under the assumption that inscribed and circumscribed ellipses that best encompass the boundary of the area were built as regard to a restricted convex domain of variation of the second-order derivative. Thus, bilateral evaluation of the derivative norm of restricted complex-valued functions with a convex domain of variation of the second-order derivative is expressed through the norm of the function and size of the ellipses covering the boundary of a convex domain.