Recently, some novel inequalities have been proposed such as the auxiliary function-based integral inequality and the Bessel-Legendre inequality which can be obtained from the former by choosing Legendre polynomials as auxiliary functions. These inequalities have been successfully applied to systems with constant delays but there have been some difficulties in application to systems with time-varying delays since the resulting bounds contain the reciprocal convexity which may not be tractable as it is. This paper proposes an equivalent form of the Bessel-Legendre inequality, which has the advantage of being easily applied to systems with time-varying delays without the reciprocal convexity. (C) 2018 Published by Elsevier Ltd.