The Bojanov – Naidenov problem for functions with nonsymmetric restrictions on the highest derivative - Kofanov, V. A. (Ukrains’kyi Matematychnyi Zhurnal, 2019)
For given r , p, α , β , μ > 0, we solve the extreme problems ^b_ax^q_± (t)dt → sup, q ≥ p, in the set of pairs (x, I) of functions x L^r_∞ and intervals I = [a, b] R satisfying the inequalities β ≤ x(r)(t) ≤ α for almost all t R , the conditions L(x_±)p ≤ L(( φ^α ,β_λ ,r) )_p, and the corresponding condition μ(sup p_[a,b]x_+) ≤ μ or μ ( sup p_[a,b]x ) ≤ μ, where L(x)p := sup { \| x\| L_p[a,b] : a, b R , | x(t)| > 0, t (a, b)}, sup p_[a,b]x_± := { t [a, b] : x_± (t) > 0} , φ^α ,β_λ ,r is the nonsymmetric (2π /λ)-periodic Euler spline of order r. As a consequence, we solve the same problems for the intermediate derivatives x(k)_± , k = 1,..., r_1, with q ≥ 1.