Mixed A_p-A_∞ estimates with one supremum

We establish several mixed A_p-A_∞ bounds for Calderón-Zygmund operators that only involve one supremum. We address both cases when the A1 part of the constant is measured using the exponential-logarithmic definition and using the Fujii-Wilson definition. In particular, we answer a question of the first author and provide an answer, up to a logarithmic factor, to a conjecture of Hytönen and Lacey. Moreover, we give an example to show that our bounds with the logarithmic factors can be arbitrarily smaller than the previously known bounds (both with one supremum and two suprema).