A new sharpened version of the Neuberg–Pedoe inequality

In this work, we present a new sharpened version of the classical Neuberg–Pedoe inequality. As an application, the following improved Neuberg–Pedoe inequality is derived: a12θ(b22θ+c22θ−a22θ)+b12θ(c22θ+a22θ−b22θ)+c12θ(a22θ+b22θ−c22θ)≥3(16/3)θF1θF2θ+2(a1θb2θ−a2θb1θ)2+2(a1θb1θ(b2θ+c2θ−a2θ)(c2θ+a2θ−b2θ)−a2θb2θ(b1θ+c1θ−a1θ)(c1θ+a1θ−b1θ))2.