Some Inequalities via Functional Type Generalization of Cauchy-Bunyakovsky- Schwarz Inequality

The Cauchy-Bunyakovsky-Schwarz inequality and its various refinements are very important in mathematical analysis. In this work, we first introduce an inequality of the form $$\left[f^{(n)}(x)\right]^2 \leq k(x) \sum_{k=0}^m a_k f^{(m-k)}\left(\frac{p}{r} x+q\right) \sum_{k=0}^l b_k f^{(l-k)}\left(\left(\frac{2}{r}-\frac{p}{r}\right) x-q\right)$$ and by using a functional type generalization of the Cauchy-Bunyakovsky-Schwarz inequality we get some inequalities for derivatives of a one-parameter deformation of the Gamma function to satisfy the introduced inequality. Also, we show that the established results are generalizations of some previous results.