On the interpolation constants for variable Lebesgue spaces

For (Figure presented.) and variable exponents (Figure presented.) and (Figure presented.) with values in [1, ∞], let the variable exponents (Figure presented.) be defined by (Figure presented.) The Riesz–Thorin–type interpolation theorem for variable Lebesgue spaces says that if a linear operator T acts boundedly from the variable Lebesgue space (Figure presented.) to the variable Lebesgue space (Figure presented.) for (Figure presented.), then (Figure presented.) where C is an interpolation constant independent of T. We consider two different modulars (Figure presented.) and (Figure presented.) generating variable Lebesgue spaces and give upper estimates for the corresponding interpolation constants C _max and C _sum, which imply that (Figure presented.) and (Figure presented.), as well as, lead to sufficient conditions for (Figure presented.) and (Figure presented.). We also construct an example showing that, in many cases, our upper estimates are sharp and the interpolation constant is greater than one, even if one requires that (Figure presented.), (Figure presented.) are Lipschitz continuous and bounded away from one and infinity (in this case, (Figure presented.)).