Bounds onKl3-Decay Form Factors

We rederive certain bounds on $K_{l3\mathrm{}}$ decay parameters found recently by Li, Pagels, and Okubo and obtain some new results with a simpler method based on a direct application of the maximum-modulus theorem for holomorphic functions. For an arbitrary value of the momentum-transfer variable $t$ in the complex cut $t$ plane, we find that the domain of values which can be taken by the form factor $d\left(t\right)\mathrm{}$ of the divergence of the weak strangeness-changing vector current $V_{\mu }^{\mathrm{}\left(K\right)\mathrm{}}$ is bounded by a circle in the plane ($\mathrm{Re}d\left(t\right),\mathrm{Im}d\left(t\right)\mathrm{}$). We express the radius and the position of the center of this circle in terms of $f_{+}\mathrm{}\left(0\right)\mathrm{}$ and of the propagator $\Delta \mathrm{}\left(t\right)\mathrm{}$ of the divergence of $V_{\mu }^{\mathrm{}\left(K\right)\mathrm{}}$.