An order relation for (\(\psi\) ,\({\bar \beta}\))-derivatives (Q1093831)

Let \(a_ k\), \(b_ k\) be the Fourier coefficients of \(f\in L(0,2\pi)\) and suppose that for arbitrary \(\psi\) :\(\beta\) : \({\mathbb{N}}\to {\mathbb{R}}\), \[ \sum^{\infty}_{k=1}(\psi (x))^{-1}(a_ k\cos (kx+\beta (k)\pi /2)+b_ k\sin (kx+\beta (k)\pi /2) \] is the Fourier series of some function in L(0,2\(\pi)\). This function is the (\(\psi\),\(\beta)\)-derivative of f and is denoted by \(f^{\psi}_{\beta}\). The set of functions f which satisfy these conditions is denoted by \(L^{\psi}_{\beta}\). For given pairs of sequences \((\psi_ i,\beta_ i)\) \((i=1,2)\), it is said that the pair \((\psi_ 1,\beta_ 1)\) L-precedes the pair \((\psi_ 2,\beta_ 2)\) if \(L^{\psi_ 2}_{\beta_ 2}\subseteq L^{\psi_ 1}_{\beta_ 1}\), in symbols \((\psi_ 1,\beta_ 1)\leq^{L}(\psi_ 2,\beta_ 2)\). The author proves: If \(f\in L^{\psi_ 2}_{\beta_ 2}\) and \((\psi_ 1,\beta_ 1)\leq^{L}(\psi_ 2,\beta_ 2)\), then there exists the derivative \(f^{\psi_ 1}_{\beta_ 1}\) which lies in \(L^{\psi}_{\beta}\), where \(\psi (k)=\psi_ 2(k)/\psi_ 1(k)\) and \(\beta (k)=\beta_ 2(k)-\beta_ 1(k)\).