On the degree of complex rational approximation to real functions

Iff∈C[?1, 1] is real-valued, letE _R ^mn (f) andE _C ^mn (f) be the errors in best approximation tof in the supremum norm by rational functions of type (m, n) with real and complex coefficients, respectively. We show that form≥n?1≥0 $$gamma _{mn} = inf { {{E_{mn}^C (f)} mathord{left/ {vphantom {{E_{mn}^C (f)} {E_{mn}^R (f)}}} right. kern-nulldelimiterspace} {E_{mn}^R (f)}}:f in C[ - 1,1]} = tfrac{1}{2}.$$