MONOTONICITY OF SEQUENCES INVOLVING GEOMETRIC MEANS OF POSITIVE SEQUENCES WITH LOGARITHMICAL CONVEXITY

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1 MONOTONICITY OF SEQUENCES INVOLVING GEOMETRIC MEANS OF POSITIVE SEQUENCES WITH LOGARITHMICAL CONVEXITY FENG QI AND BAI-NI GUO Abstract. Let f be a positive fuctio such that x [ f(x + )/f(x) ] is icreasig o [, ), the the sequece { / f() } is de- = creasig. If f is a logarithmically cocave ad positive fuctio defied o [, ), the the sequece { / f() } is icreasig. = As cosequeces of these mootoicities, the lower ad upper bouds for +k / the ratio +m +k+m of the geometric mea sequece { } +k are obtaied, where k is a oegative iteger ad m = a atural umber. Some applicatios are give.. Itroductio It is kow that, for N, the followig double iequality were give i [6]: which ca be reaaraged as! < <, () + ()! [Γ( + r)] r < [Γ( + r)] r+ () ad [Γ( + r)] r r > [Γ( + r)] r+. (3) r Mathematics Subject Classificatio. Primary 6D5; Secodary 6A48. Key words ad phrases. mootoicity, iequality, geometric mea, ratio, positive sequece, logarithmically cocave, mathematical iductio. The authors were supported i part by NNSF (#00006) of Chia, SF for the Promiet Youth of Hea Provice (#000000), SF of Hea Iovatio Talets at Uiversities, NSF of Hea Provice (# ), Chia. This paper was typeset usig AMS-LATEX.

2 F. QI AND B.-N. GUO I [], the left iequality i () was refied by ( < / ) + /r i r i r <! + ()! (4) for all positive real umbers r. Both bouds are the best possible. Usig aalytic method ad Stirlig s formula, i [0, 4, 6, 7], for, m N ad k beig a oegative iteger, the author ad others proved the followig iequalities: ( +k ) / /( + k + +m+k /(+m) + k + m + k + < i i) + m + k, (5) the equality i (5) is valid for = ad m =, which exted ad refie those i (). There is a rich literature o refiemets, extesios, ad geeralizatios of the iequalities i (4), for examples, [, 8, 9, 3, 9] ad refereces therei. Note that the iequalities i (4) are direct cosequeces of a cojecture which states that the fuctio ( ir/ + + ir) /r is decreasig with r. Please refer to [8]. I [], usig the ideas ad method i [3, 5, 5] ad the mathematical iductio, the followig iequalities were obtaied. Theorem A. Let k be a oegative iteger, ad m positive itegers, ad α [0, ] a costat. The + k + + α + m + k + + α < [ +k ] / (i + α) + k + α ] /(+m) (i + α) + m + k + α. (6) [ +m+k If = ad m =, the the equality i the right had side iequality of (6) holds. I [], Theorem A was geeralized to the followig Theorem B. For all oegative itegers k ad atural umbers ad m, we have a( + k + ) + b a( + m + k + ) + b < [ +k ] (ai + b) [ +m+k ] (ai + b) +m a( + k) + b a( + m + k) + b, (7) where a is a positive costat, ad b is a oegative costat. The equality i (7) is valid for = ad m =.

3 MONOTONICITY OF SEQUENCES INVOLVING GEOMETRIC MEANS 3 I [4], the followig mootoicity results for the gamma fuctio were established: The fuctio [Γ( + x )]x decreases with x > 0 ad x[γ( + x )]x icreases with x > 0, which recover the iequalities i () which refer to iteger values of r. These are equivalet to the fuctio [Γ( + x)] x beig icreasig ad [Γ(+x)] x x beig decreasig o (0, ), respectively. I additio, it was proved that the fuctio x γ [Γ( + x )x ] decreases for 0 < x <, where γ = deotes the Euler s costat, which is equivalet to [Γ(+x)] x x γ beig icreasig o (, ). I [4], the followig mootoicity result was obtaied: The fuctio [Γ(x + y + )/Γ(y + )] /x x + y + (8) is decreasig i x for fixed y 0. The, for positive real umbers x ad y, we have x + y + x + y + [Γ(x + y + )/Γ(y + )]/x. (9) [Γ(x + y + )/Γ(y + )] /(x+) Iequality (9) exteds ad geeralizes iequality (5), sice Γ() =!. Defiitio ([7, p. 7]). A positive fuctio f : I R, I a iterval i R, is said to be logarithmically covex (log-covex, multiplicatively covex) if l f is covex, or equivaletly if for all x, y I ad all α [0, ], f(αx + ( α)y) f α (x)f α (y). (0) It is said to be logarithmically cocave (log-cocave) if the iequality i (0) is reversed. Remark. By f = exp l f, it follows that a logarithmically covex fuctio is covex (but ot coversely). This directly follows from (0), of course, sice by the arithmetic-geometric iequality we have f α (x)f α (y) αf(x) + ( α)f(y). J. Pečarić told the author that a cocave positive fuctio is a logarithmically cocave oe affirmatively. I this article, we will further geeralize the iequlaities i (7) ad obtai the followig

4 4 F. QI AND B.-N. GUO Theorem. Let f be a icreasig, logarithmically covex ad positive fuctio defied o [, ). The the sequece { f() } = is decreasig. As a cosequece, we have the followig +k +m +m+k where m is a atural umber ad k a oegative iteger. Corollary. Let {a i } sequece, the the sequece () f( + k + ) f( + m + k + ), () be a icreasig, logarithmically covex, ad positive { a! a + } = is decreasig. As a cosequece, we have the followig (3) a! +m a +m! a + a +m+, (4) where m is a atural umber ad a! is the sequece factorial defied by a i. Theorem. Let f be a logarithmically cocave ad positive fuctio defied o [, ). The the sequece { } f() = is icreasig. As a cosequece, we have the followig +k f( + k) +m +m+k f( + m + k), (6) where m is a atural umber ad k a oegative iteger. The equality i (6) is valid for = ad m =. Corollary. Let {a i } be a logarithmically cocave ad positive sequece. The the sequece { } a! (7) a = is icreasig. Therefore, we have a! +m a +m! a, (8) a +m (5)

5 MONOTONICITY OF SEQUENCES INVOLVING GEOMETRIC MEANS 5 where m is a atural umber ad a! is the sequece factorial defied by a i. The equality i (8) is valid for = ad m =. At last, i Sectio 3, some applicatios of Theorem ad Theorem are give ad a ope problem is proposed. Remark. It is well kow that the left had side term i () or (6) is a ratio of two geometric meas of sequece {}.. Proofs of Theorem ad Theorem Proof of Theorem. The mootoicity of the sequece () ad iequality () are equivalet to the followig ( ) / ( + ) /(+), f() f( + ) + l l f() + f( + ) l l f( + ), f(). (9) Sice l x is cocave o (0, ), by defiitio of cocaveess, it follows that, for i, i f(i + ) l f( + ) + i + l ( i f(i + ) l f( + ) + i + ( ) if(i + ) + ( i + ) = l. ()f( + ) f( + ) f( + ) ) (0) Sice f is logarithmically covex, we have f()f( + ) [f()]. Hece, for all i, from the fuctio f beig icreasig, we have f()f( + ) [f()] f()[f() f( + )] ()f( + ) f() ()f( + ) f() ()f( + ) f() () () f() f() f() f() if(i + ) () i

6 6 F. QI AND B.-N. GUO if(i + ) + ( i + ) if(i + ) + ( i + ) ()f( + ) ()f( + ) f() f(). Combiig the last lie above with (0) yields i f(i + ) l f( + ) + i + l f( + ) l f(). () Summig up o both sides of () from to ad simplifyig reveals iequality (9). The proof is complete. Proof of Theorem. The mootoicity of the sequece (5) ad iequality (6) are equivalet to the followig f() + + f() Suppose iequality (3) is valid for some >. Sice, by the iductive hypothesis l + l [ ] l f() l f() ( + ) + l l [ ] l f() l f() l f() For =, the equality i (3) holds. l f() l. (3) + l = = [ ] l + by iductio, it is sufficiet to prove l f() [ l f() l f() l f() f(), ] + l f() l f() l f() + l f() l f( + ) l f() l f() l f( + ) l[f()f( + )] l f ()

7 MONOTONICITY OF SEQUENCES INVOLVING GEOMETRIC MEANS 7 f()f( + ) f (), this follows from the logarithmic cocaveess of the fuctio f. The proof is complete. Remark 3. If the fuctio f i Theorem is differetiable, the we ca give the followig proof of Theorem by Cauchy s mea value theorem ad mathematical iductio. Proof of Theorem uder coditio such that f beig differetiable. The mootoicity of the sequece () ad iequality () are equivalet to l + l l f() l f( + ) l l f() () [ l f() l f( + ) ] ( + ) l f() () l f( + ) l. (4) For =, iequality (4) ca be rewritte as f()[f(3)] [f()] 3. Sice f is logarithmically covex ad icreasig, we have f()f(3) [f()] ad f(3) f(), respectively. Therefore, iequality (4) holds for =. Suppose iequality (4) is valid for some >. The, by iductive hypothesis, we have + l = [ ] f() l + [ ] f() ( + ) l f() () l f( + ) + = () l f() l f( + ). hece, by iductio, it is sufficiet to prove the followig () l f() l f( + ) ( + 3) l f( + ) ( + ) l f( + 3), which ca be rearraged as ()[l f() l f( + )] ( + )[l f( + ) l f( + 3)],

8 8 F. QI AND B.-N. GUO further, sice f is icreasig, l f( + ) l f() l f( + 3) l f( + ) +. (5) Usig Cauchy s mea values applied to g(x) = l f( + x) ad h(x) = l f( + + x) for x [0, ] i iequality (5), it follows that there exists a poit ξ (0, ) such that f ( + ξ) f( + ξ) f( + + ξ) f ( + + ξ) +. Sice the positive fuctio f is logarithmically covex ad differetiable, the [l f(x)] = f (x) f(x) is icreasig. Thus f ( + ξ) f( + ξ) f ( + + ξ) f( + + ξ), ad the f ( + ξ) f( + ξ) f( + + ξ) f ( + + ξ) < +. Iequality (5) follows. The proof is complete. 3. Applicatios 3.. The affie fuctio f(x) = ax + b for x > b a, where a > 0 ad b R are costats, is positive ad logarithmically cocave. From Theorem applied to this affie fuctio, the right had side iequality i (7) follows. 3.. From procedure of the proof of Theorem ad oticig iequality (), we ca establish the followig more geeral results. Theorem 3. Let f be a positive fuctio such that x [ f(x+) f(x) [, ), the the sequece () decreases ad iequality () holds. Corollary 3. Let {a i } be a positive sequece such that { i [ a i+ a i icreasig, the the sequece (3) decreases ad iequality (4) holds. ] is icreasig o ]} is 3.3. The left had side iequality i (7) follows from Corollary 3.

9 MONOTONICITY OF SEQUENCES INVOLVING GEOMETRIC MEANS Applyig Theorem 3 or Corollary 3 to f(x) = Γ(x + ) or a i = i! respectively yields i= + i (i + k) +m +m+ i i= (i + k) +m Similarly, we have = +k (i!) +m +m+k (i!) ( + k + )! ( + m + k + )! = (6) ( + k + + i). m +k (i!!) +m +m+k +k ((i)!!) +m +m+k +k +m +m+k (i!!) ((i)!!) ((i )!!) ((i )!!) ( + k + )!! ( + m + k + )!!, (7) [( + k + )]!! [( + m + k + )]!!, (8) [( + k) + ]!! [( + m + k) + ]!!. (9) Where ad m are atural umbers ad k a oegative iteger I Corollary, cosiderig the sequece {l a i } is icreasig, covex, ad positive, we obtai the followig Corollary 4. Let {a i } be a icreasig covex positive sequece ad A = a i a arithmetic mea. The the sequece A a + decreases. This gives a lower boud for differece of two arithmetic meas: where m is a atural umber. A A +m a + a +m+, (30) 3.6. I Corollary, cosiderig the sequece {l a i } we have is cocave ad positive, Corollary 5. Let {a i } be a cocave positive sequece ad A = a i a arithmetic mea. The the sequece A a icreases. This implies a upper boud for differece of two aithmetic meas: where m is a atural umber. A A +m a a +m, (3)

10 0 F. QI AND B.-N. GUO 3.7. For real umbers b ad c 0 such that b > c, the fuctio x + bx + c is logarithmically cocave ad satisfies coditios of Theorem 3, the we have ( + k + ) + b( + k + ) + c ( + m + k + ) + b( + m + k + ) + c +k (i + bi + c) +m +m+k where m is a atural umber ad k a oegative iteger. (i + bi + c) ( + k) + b( + k) + c ( + m + k) + b( + m + k) + c, (3) 4. Ope Problem I the fial, we pose the followig ope problem. Ope Problem. For ay positive real umber z, defie z! = z(z ) {z}, where {z} = z [z ], ad [z] deotes Gauss fuctio whose value is the largest iteger ot more tha z. Let x > 0 ad y 0 be real umbers, the x + x x! x x + y + x+y (x + y)! x + y. (33) Refereces [] H. Alzer, O a iequality of H. Mic ad L. Sathre, J. Math. Aal. Appl. 79 (993), [] T. H. Cha, P. Gao, ad F. Qi, O a geeralizatio of Martis iequality, RGMIA Res. Rep. Coll. 4 (00), o., Art., Available olie at v4.html. [3] Ch.-P. Che, F. Qi, P. Ceroe, ad S. S. Dragomir, Mootoicity of sequeces ivolvig covex ad cocave fuctios, RGMIA Res. Rep. Coll. 5 (00), o., Art.. Available olie at [4] D. Kershaw ad A. Laforgia, Mootoicity results for the gamma fuctio, Atti Accad. Sci. Torio Cl. Sci. Fis. Mat. Natur. 9 (985), [5] J.-Ch. Kuag, Some extesios ad refiemets of Mic-Sathre iequality, Math. Gaz. 83 (999), 3 7. [6] H. Mic ad L. Sathre, Some iequalities ivolvig (r!) /r, Proc. Ediburgh Math. Soc. 4 (964/65), 4 46.

11 MONOTONICITY OF SEQUENCES INVOLVING GEOMETRIC MEANS [7] J. Pečarić, F. Proscha, ad Y. L. Tog, Covex Fuctios, Partial Orderigs, ad Statistical Applicatios, Mathematics i Sciece ad Egieerig 87, Academic Press, [8] F. Qi, A algebraic iequality, J. Iequal. Pure Appl. Math. (00), o., Art. 3. Available olie at RGMIA Res. Rep. Coll. (999), o., Art. 8, Available olie at [9] F. Qi, Geeralizatio of H. Alzer s iequality, J. Math. Aal. Appl. 40 (999), [0] F. Qi, Iequalities ad mootoicity of sequeces ivolvig ( + k)!/k!, RGMIA Res. Rep. Coll. (999), o. 5, Art. 8, Available olie at html. [] F. Qi, Iequalities ad mootoicity of the ratio for the geometric meas of a positive arithmetic sequece with uit differece, submitted. [] F. Qi, Iequalities ad mootoicity of the ratio for the geometric meas of a positive arithmetic sequece with arbitrary differece, Tamkag. J. Math. 34 (003), o. 3, accepted. [3] F. Qi ad B.-N. Guo, A iequality betwee ratio of the exteded logarithmic meas ad ratio of the expoetial meas, Taiwaese J. Math. (003), accepted. RGMIA Res. Rep. Coll. 4 (00), o., Art. 8, Available olie at [4] F. Qi ad B.-N. Guo, Iequalities ad mootoicity for the ratio of gamma fuctios, Taiwaese J. Math. (003), accepted., 3 [5] F. Qi ad B.-N. Guo, Mootoicity of sequeces ivolvig covex fuctio ad sequece, RGMIA Res. Rep. Coll. 3 (000), o., Art. 4, Available olie at vu.edu.au/v3.html. [6] F. Qi ad B.-N. Guo, Some iequalities ivolvig the geometric mea of atural umbers ad the ratio of gamma fuctios, RGMIA Res. Rep. Coll. 4 (00), o., Art. 6, Available olie at [7] F. Qi ad Q.-M. Luo, Geeralizatio of H. Mic ad J. Sathre s iequality, Tamkag J. Math. 3 (000), o., RGMIA Res. Rep. Coll. (999), o. 6, Art. 4, Available olie at [8] J. A. Sampaio Martis, Iequalities of Rado-Popoviciu type, I: Marques de Sá, Eduardo (ed.) et al. Mathematical studies. Homage to Professor Doctor Luís de Albuquerque. Coimbra: Uiversidade de Coimbra, Faculdade de Ciêcias e Tecologia, Departameto de Matemática, (994). [9] N. Towghi ad F. Qi, A iequality for the ratios of the arithmetic meas of fuctios with a positive parameter, RGMIA Res. Rep. Coll. 4 (00), o., Art. 5, Available olie at

12 F. QI AND B.-N. GUO (Qi ad Guo) Departmet of Applied Mathematics ad Iformatics, Jiaozuo Istitute of Techology, #4, Mid-Jiefag Road, Jiaozuo City, Hea , Chia address, Qi: URL, Qi: address, Guo: