Let ƒ be a transcendental entire function. The fast escaping set, A(ƒ), plays a key role in transcendental dynamics. The quite fast escaping set, Q(ƒ), defined by an apparently weaker condition is equal to A(ƒ) under certain conditions. Here we introduce Q2(ƒ) defined by what appears to be an even weaker condition. Using a new regularity condition we show that functions of finite order and positive lower order satisfy Q2(ƒ) = A(ƒ). We also show that the finite composition of such functions satisfies Q2(ƒ) = A(ƒ). Finally, we construct a function for which Q2(ƒ) ≠ Q(ƒ) = A(ƒ)
