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We obtain stringent bounds in the  < r^2 >_S^{K pi}-c plane where these are the scalar radius and the curvatureparameters of the scalar pi K form factor respectively using analyticity anddispersion relation constraints, the knowledge of the form factor from thewell-known Callan-Treiman point m_K^2-m_pi^2, as well as at m_pi^2-m_K^2 whichwe call the second Callan-Treiman point. The central values of these parametersfrom a recent determination are accomodated in the allowed region provided thehigher loop corrections to the value of the form factor at the secondCallan-Treiman point reduce the one-loop result by about 3% with F_K/F_pi=1.21.Such a variation in magnitude at the second Callan-Treiman point yields 0.12fm^2 \lesssim < r^2>_S^{K pi} \lesssim 0.21 fm^2 and 0.56 GeV^{-4} \lesssim c\lesssim 1.47 GeV^{-4} and a strong correlation between them. A smaller valueof F_K/F_pi shifts both bounds to lower values.

Improved bounds on the radius and curvature of the K pi scalar form factor and implications to low-energy theorems