2024/05/30 by Nayak, Raj Kumar
#47A12 #47A30 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2405.19698
In this article, we establish an improvement of the Cauchy-Schwarz inequality. Let x, y ∈ H, and let f: (0,1) → ℝ+ be a well-defined function, where ℝ+ denote the set of all positive real numbers. Then |⟨ x, y ⟩|2 ≤ (f(t))/(1+f(t)) ‖x‖2 ‖y‖2 + (1)/(1+ f(t)) |⟨ x, y ⟩ | ‖x‖‖y‖. We have applied this result to derive new and improved upper bounds for the numerical radius.