Concavity and bounds involving generalized elliptic integral of the first kind

. In the article, we provide a suf fi cient condition for value range of the constant c such that the function x → K a ( √ x ) / log ( c / √ 1 − x ) is strictly concave on ( 0 , 1 ) for a ∈ ( 0 , 1 / 2 ] , which generalize a very recently obtained result that the function x → K ( √ x ) / log ( c / √ 1 − x ) is strictly concave on ( 0 , 1 ) if and only if c = e 4 / 3 . As applications, we present new bounds for K a ( x ) , K a ( √ 1 − x 2 ) / K a ( √ x ) and K a ( √ 1 − x 2 ) K a ( √ x ) , where K a ( x ) is the generalized elliptic integral of the fi rst kind and K ( x ) = K 1 / 2 ( x ) .