On the interpolation constants for variable Lebesgue spaces

For theta is an element of (0,1) and variable exponents p(0)(center dot), q(0)(center dot) and p(1)(center dot), q(1)(center dot) with values in [1, infinity], let the variable exponents p(theta)(center dot), q(theta)(center dot) be defined by 1/p(theta)(center dot) := (1 - theta)/p(0)(center dot) + theta/p(1)(center dot), 1/q(theta) (center dot) := (1 - theta)/q(0)(center dot) + theta/q(1)(center dot). The Riesz-Thorin-type interpolation theorem for variable Lebesgue spaces says that if a linear operator.. acts boundedly fromthe variable Lebesgue space L-pj (center dot) to the variable Lebesgue space L-pj (center dot) for j = 0,1, then parallel to T parallel to(Lp theta(center dot)) -> (Lq theta(center dot)) <= C parallel to T parallel to(Lp theta(center dot)) -> (1-theta)(Lq theta(center dot)) parallel to T parallel to(Lp1(center dot)) -> (theta)(Lq1(center dot)) where C is an interpolation constant independent of T. We consider two different modulars e(max)(center dot) and e(sum)(center dot) generating variable Lebesgue spaces and give upper estimates for the corresponding interpolation constants C-max and C-sum, which imply that C-max <= 2 and C-sum <= 4, as well as, lead to sufficient conditions fo C-max = 1 and C-sum = 1. 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 p(j)(center dot) = qj(center dot), j =0,1 are Lipschitz continuous and bounded away from one and infinity (in this case, e(max)(center dot) = e(sum)(center dot)).