In 1973, Arimoto proved the strong converse theorem for the discrete memoryless channels stating that when transmission rate R is above channel capacity C, the error probability of decoding goes to one as the block length n of code word tends to infinity. He proved the theorem by deriving the exponent function of error probability of correct decoding that is positive if and only if R > C. Subsequently, in 1979, Dueck and Körner determined the optimal exponent of correct decoding. Recently the author determined the optimal exponent on the correct probability of decoding have the form similar to that of Dueck and Körner determined. In this paper we give a rigorous proof of the equivalence of the above exponet function of Dueck and Körner to a exponent function which can be regarded as an extention of Arimoto's bound to the case with the cost constraint on the channel input.