We prove that Gadjieva's conjecture holds true as stated in her PhD thesis. The positive solution of this conjecture allows us to obtain improved versions of the Jackson--Stechkin type inequalities obtained in her thesis and some others. As an application, an equivalence of the modulus of smoothness with the realization functional is established. We obtain a characterization class for the modulus of smoothness.
"Gadjieva's conjecture, $K$-functionals, and someapplications in weighted Lebesgue spaces,"
Turkish Journal of Mathematics: Vol. 42:
3, Article 58.
Available at: https://journals.tubitak.gov.tr/math/vol42/iss3/58