− q) 1 + 8xq , 2q √ (q + q 2 ) − q(1 − q) 1 + 8xq . y2 = 2q 3 y1 = Since y1 ≤ 1+q 2q , y2 ≤ 1+q 2q 2 and q k < 1+q 2q < 1+q 2q 2 , from (4), it follows that f (q k ) ≤ f (y1 ) = 0 = g(y2 ) < g(q k ). Thus y1 < q k < y2 and hence 1 − qk 1 − y1 1 − y2 < < . 1−q 1−q 1−q i.e. √ 1 + 8xq − (1 + 2q) < Zq∗ ≤ 2q 2 √ 1 + 8xq − 1 . 2q This completes the proof. Remark. Letting q − 1− in the above theorem, we obtain S´ andor’s Theorem 1.3. We define the q-analogues of P and P∗ as follows: Pq (x) = min{m ∈ N : px ≤ Γq (m + 1)}, p > 1, x ∈ (0, ∞), Pq∗ (x) = max{m ∈ N : Γq (m + 1) ≤ px }, p > 1, x ∈ [1, ∞), and where 0 < q < 1. Clearly, Pq (x) → P (x) and Pq∗ → P∗ (x) as q → 1− . From the definitions of Pq and Pq∗ , we have Pq∗ (x) ≤ Pq (x) ≤ Pq∗ (x) + 1. Hence it is enough to study the function Pq∗ . Theorem 2.2. If 0 < q < 1, then P∗ (x) ∼ x log p ´ ³ 1 log 1−q (x → ∞).45:T542,34 T. Kim, C. Adiga and J. H. Han No. 1 Proof. If Γq (n + 1) ≤ px < Γq (n + 2), then Pq∗ (x) = n and log Γq (n + 1) ≤ log px < log Γq (n + 2). But by the q-analogue of Stirling’s formula established by Moak [4], we have µ ¶ µ n+1 ¶ µ ¶ 1 q 1 log Γq (n + 1) ∼ n + log ∼ n log . 2 q−1 1−q ³ ´ 1 Dividing (5) throughout by n log 1−q , we obtain log Γq (n + 1) x log p log Γq (n + 2) ³ ´ ≤ ´< ³ ´. ³ 1 1 1 n log 1−q Pq∗ (x) log 1−q n log 1−q (5) (6) (7) Using (6) and (7), we deduce lim x→∞ x log p ³ Pq∗ (x) log 1 1−q ´ = 1. This completes the proof. References [1] C. Adiga and T. Kim, On a generalization of S´ andor’s function, Proc. Jangjeon Math. Soc., 5(2002), 121-124. [2] C. Adiga, T. Kim, D. D. Somashekara and N. Fathima, On a q-analogue of S´ andor’s function, J. Inequal. Pure and Appl. Math., 4(2003), 1-5. [3] T. Kim, Non-archimedean q-integrals with multiple Changhee q-Bernoulli polynomials, Russian J. Math. Phys., 10(2003), 91-98. [4] D. S. Moak, The q-a