ON MONOTONICITY OF SOME COMBINATORIAL SEQUENCES

Transcription

1 Publ. Math. Debrece 8504, o. 3-4, ON MONOTONICITY OF SOME COMBINATORIAL SEQUENCES QING-HU HOU*, ZHI-WEI SUN** AND HAOMIN WEN Abstract. We cofirm Su s cojecture that F / F 4 is strictly decreasig to the limit, where F 0 is the Fiboacci sequece. We also prove that the sequece D / D 3 is strictly decreasig with limit, where D is the -th deragemet umber. For m-th order harmoic umbers H we show that H / H = = /m =,, 3,..., 3 is strictly icreasig.. Itroductio A challegig cojecture of Firoozbaht states that p > p for every =,, 3,..., where p deotes the -th prime. Note that lim p = by the Prime Number Theorem. I [4] the secod author cojectured further that for ay iteger > 4 we have the iequality p p < log log, which has bee verified for all Motivated by this ad [3], Su [4, Coj..] cojectured that the sequece S / S 7 is strictly icreasig, where S is the sum of the first positive squarefree umbers. Moreover, he also posed may cojectures o mootoicity of sequeces of the type a / a N with a a familiar combiatorial sequece of positive itegers. Throughout this paper, we set N = {0,,,...} ad Z = {,, 3,...}. Let A ad B be itegers with = A 4B 0. The Lucas sequece u = u A, B N is defied as follows: u 0 = 0, u =, ad u = Au Bu for =,, 3,.... Key words ad phrases. Combiatorial sequeces, mootoicity, log-cocavity 00 Mathematics Subject Classificatio. Primary 05A0; Secodary B39, B75. *Supported by the Natioal Natural Sciece Foudatio grat 767 of Chia. **Supported by the Natioal Natural Sciece Foudatio grat 740 of Chia.

2 QING-HU HOU, ZHI-WEI SUN AND HAO-MIN WEN It is well ow that u = α β /α β for all N, where α = A ad β = A are the two roots of the characteristic equatio x Ax B = 0. The sequece F = u, N is the famous Fiboacci sequece, see [, p. 46] for combiatorial iterpretatios of Fiboacci umbers. Our first result is as follows. Theorem.. Let A > 0 ad B 0 be itegers with = A 4B > 0, ad set u = u A, B for N. The there exists a iteger N > 0 such that the sequece u / u N is strictly decreasig with limit. I the case A = ad B = we may tae N = 4. Remar.. Uder the coditio of Theorem., by [, Lemma 4] we have u < u uless A = =. Note that the secod assertio i Theorem. cofirms a cojecture of the secod author [4, Coj. 3.] o the Fiboacci sequece. For Z the th deragemet umber D deotes the umber of permutatios σ of {,..., } with σi = i for o i =,...,. It has the followig explicit expressio cf. [, p. 67]: D =!!. =0 Our secod theorem is the followig result cojectured by the secod author [4, Coj. 3.3]. Theorem.. The sequece D / D 3 with limit. is strictly decreasig Remar.. It follows from Theorem. that the sequece D is strictly icreasig. For each m Z those H = = /m Z are called harmoic umbers of order m. The usual harmoic umbers of order are those ratioal umbers H = H =,, 3,.... Our followig theorem cofirms Cojecture.6 of Su [4]. Theorem.3. For ay positive iteger m, the sequece is strictly icreasig. H / H 3

3 ON MONOTONICITY OF SOME COMBINATORIAL SEQUENCES 3 We will prove Theorems..3 i Sectios 4 respectively. It seems that there is o simple form for the geeratig fuctio =0 a x with a = u, D, H. Note also that the set of those sequeces a of positive umbers with a / a decreasig or icreasig is closed uder multiplicatio. Proof of Theorem.. Set The α = A log u = log α γ α β for ay Z. Note that log u = log u u log u Sice we deduce that log γ lim. Proof of Theorem., β = A, ad γ = β α = A A. γ = lim = log α log γ log = log γ log γ log. = 0 ad lim = 0, lim log u = 0, i.e., lim u u u =. For ay Z, clearly u > u log u u u log u Observe that = log α log γ log α log γ := log u log log γ > log u log u log u log log = log γ log γ log u > 0. log log γ.

4 4 QING-HU HOU, ZHI-WEI SUN AND HAO-MIN WEN The fuctio fx = log x o the iterval, is cocave sice f x = /x < 0. Note that γ <. If γ x 0, the t = x/ γ [0, ] ad hece fx = ft γ t0 tf γ tf0 = qx, where q = log γ / γ > 0. Note also that log x < x for x > 0. So we have log γ log γ q γ, log γ log γ < γ, log γ log γ < γ. Therefore ad hece > log q γ γ γ > log γ q γ γ. Sice lim γ = 0, whe > we have > 0 for large. Now it remais to cosider the case =. Clearly γ = A /A > 0. Recall that log x < x for x 0,. As d dx log xxx = x x x = x x > 0 for x 0, 0.5, we have log x x x > log 0 0 = 0 for x 0, 0.5. If is large eough, the γ < 0.5 ad hece where = log γ log γ log γ > w, w := γ γ γ γ. Note that w lim γ = γ γ = γ > 0. So, for sufficietly large we have > w > 0. Now we show that 4 suffices i the case A = ad B =. Note that = 5 ad γ The sequece γ is

5 ON MONOTONICITY OF SOME COMBINATORIAL SEQUENCES 5 decreasig sice 3 γ < for. It follows that γ γ < /3 for 6. I view of, if 6 the > log 5 γ q γ γ > log 5 γ 3 > log 5 > 0. It is easy to verify that 4 ad 5 are positive. So F / F 4 is strictly decreasig. I view of the above, we have completed the proof of Theorem.. 3. Proof of Theorem. Proof of Theorem.. Let 3. It is well ow that D!/e / cf. [, p. 67]. Applyig the Itermediate Value Theorem i calculus, we obtai log D log! e D! e 0.5. Set R 0 = log D log!. The R 0.5. Sice lim R 0 / = 0, we have log D lim log D log! = lim log! log log! log! = lim log log / log! = lim log /! = lim. As! π/e i.e., lim!/ π/e = by Stirlig s formula, we have log /! ad hece log D lim log D = 0. Thus lim D / D =.

6 6 QING-HU HOU, ZHI-WEI SUN AND HAO-MIN WEN From the ow idetity D /! = =0 /!, we have the recurrece D = D for >. Thus, if 3 the R 0 R 0 = log D! log D! = log D = log. D D Fix 4. If is eve, the 0 < R 0 R 0 = log D If is odd, the 0 > R 0 R 0 = log < D = > = D D sice log x x > 0 for x 0, 0.5. So R 0 R 0 < 3 D 0.5 3e! D 3 D 0.5. D 3 D 0.5 ad hece R 0 R 0 < 3e! 3e. Similarly, we also have R 0 R 0 < Therefore, R 0 = ad hece R 0 R 0 R 0 R 0 R 0 3e! R 0 R 0 R 0 R 0 3e. R 0 6e R 0 6e 6e 3. Thus R 6e 3, where R0 R := R 0 R 0.

7 Sice we have ON MONOTONICITY OF SOME COMBINATORIAL SEQUENCES 7 log! = = < = < = log dx log xdx = log dx = log!, log xdx = log log < log! = log! log < log log ad so log! = log R with R < log. Observe that log D log D log D log! log! = log! = log log! log R R log log = R R R R = R R. If 7, the R R > log 6e 3 > 0, ad hece we get D log D > log D. D By a direct chec via computer, the last iequality also holds for = 4,..., 6. Therefore, the sequece D / D 3 is strictly decreasig. This eds the proof. Lemma 4.. For x > 0 we have 4. Proof of Theorem.3 log x > x x.

8 8 QING-HU HOU, ZHI-WEI SUN AND HAO-MIN WEN Proof. As d dx log x x x = x x, we see that log x x x / > log 0 0 / = 0 for ay x > 0. Lemma 4.. Let m, Z with 3. If m or 30, the m H log H > 4. 3 Proof. Recall that H refers to H. If 30, the H log H H 30 log H 30 > 4 ad hece 3 holds for m =. Below we assume that m. H 3 log H 3. So it suffices to show that wheever m or 30. By Lemma 4., As 3, we have H log H m H 3 log H 3 > 4 4 log H 3 = log m 3 m > m 3 m m 3 m > m 3 m m = m 3 m 4. m If m 3, the 4/3 m 4/3 3 > ad hece log H 3 > / m. Note also that H 3 log H 3 > /4. So we always have H 3 log H 3 >. m If m, the.5 m.5 > 0 ad hece > 4 m.5 4 m /, m therefore 4 holds. Whe 30, we have ad hece 4 also holds. m = 4 4m 6 6 m 4 Proof of Theorem.3. Let m ad 3. Set := log H H log H H m = log H log H log H. It suffices to show that < 0. This ca be easily verified by computer if m {,..., 0} ad {3,..., 9}.

9 ON MONOTONICITY OF SOME COMBINATORIAL SEQUENCES 9 Below we assume that m or 30. Recall ad the ow fact that log x < x for x > 0. We clearly have log H = log < H m H m H ad log H H > log It follows that = m H > m H log H H log H log H H log H < m H Sice m = m =0 m H m m H. m H. by the biomial theorem, we obtai log H m m =0 m m H Thus m log H < m m =0 m m H log H = m. m H H m H < H log H m m m <4 H log H. Applyig 3 we fid that < 0. This completes the proof. m H Acowledgmets. The iitial versio of this paper was posted to arxiv i 0 as a preprit with the ID arxiv: The authors are grateful to the two referees for their helpful commets.

10 0 QING-HU HOU, ZHI-WEI SUN AND HAO-MIN WEN Refereces [] R. P. Staley, Eumerative Combiatorics, Vol., Cambridge Uiv. Press, Cambridge, 997. [] Z.-W. Su, Reductio of uows i diophatie represetatios, Sci. Chia Ser. A 35 99, [3] Z.-W. Su, O a sequece ivolvig sums of primes, Bull. Aust. Math. Soc , [4] Z.-W. Su, Cojectures ivolvig arithmetical sequeces, i: Number Theory: Arithmetic i Shagri-La eds., S. Kaemitsu, H. Li ad J. Liu, Proc. 6th Chia- Japa Semiar Shaghai, August 5-7, 0, World Sci., Sigapore, 03, pp Qig-Hu Hou Ceter for Applied Mathematics, Tiaji Uiversity, Tiaji 30007, People s Republic of Chia address: hou@aai.edu.c Zhi-Wei Su Departmet of Mathematics, Najig Uiversity, Najig 0093, People s Republic of Chia address: zwsu@ju.edu.c Haomi We Departmet of Mathematics, The Uiversity of Pesyvaia, Philadelphia, PA 904, USA address: weh@math.upe.edu