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High-statistics data on the $e^+e^-\to \pi^+\pi^-$ cross section and the pionvector form factor have been obtained recently by several collaborations.Unfortunately, there are some tensions between different datasets, especiallythe most precise ones, which have not been resolved so far. Additionalindependent constraints on the data are therefore of interest. We consider aparametrization-free method of analytic extrapolation proposed recently, whichis based on a mixed phase and modulus extremal problem and combines rigorousupper and lower bounds with numerical simulations to account for thestatistical distributions of the input and output values. Spacelike data on theform factor and measurements of the modulus in the region $(0.65-0.71)$ GeV areused as input. In previous works, the formalism was applied for extrapolatingthe form factor to low energies. In the present work, we use it as a stringentand model-independent test of consistency with analyticity and unitarity forthe high-statistics data around the $\rho$ resonance. The study reveals someinconsistencies, in particular below the $\rho$ peak the BABAR data areslightly higher than the band of extrapolated values, while above the $\rho$peak all the data are situated at the lower edge of the band. The implicationsof the results on the two-pion vacuuum polarization contribution to theanomalous magnetic moment of the muon are briefly discussed.

Test of analyticity and unitarity for the pion form-factor data around the $\rho$ resonance