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∆ = { D if D ≡ 1 (mod 4), 4D if D 6≡ 1 (mod 4). For the case that D is an odd prime, Gut [3] proved that if D ≡ 1 (mod 4), then h < D/4. Newman [6] proved that h < 2 √ D/3. Agoh [1] proved that if ν > 1/2 and D ≡ 1 (mod 4), then h < ν√D except for a finite number of D. In this paper, we prove a general result as follows. Theorem. (a) For any square free D ∈ N, we have h ≤ [√∆/2]. (b) Moreover , if D ≡ 3 (mod 4) is an odd prime, then

Upper bounds for class numbers of real quadratic fields