Turán type inequalities for confluent hypergeometric functions of the second kind
aa r X i v : . [ m a t h . C A ] S e p TUR ´AN TYPE INEQUALITIES FOR CONFLUENT HYPERGEOMETRICFUNCTIONS OF THE SECOND KIND ´ARP ´AD BARICZ, SAMINATHAN PONNUSAMY, AND SANJEEV SINGH
Abstract.
In this paper we deduce some tight Tur´an type inequalities for Tricomi confluent hyperge-ometric functions of the second kind, which in some cases improve the existing results in the literature.We also give alternative proofs for some already established Tur´an type inequalities. Moreover, by usingthese Tur´an type inequalities, we deduce some new inequalities for Tricomi confluent hypergeometricfunctions of the second kind. The key tool in the proof of the Tur´an type inequalities is an integralrepresentation for a quotient of Tricomi confluent hypergeometric functions, which arises in the studyof the infinite divisibility of the Fisher-Snedecor F distribution. Introduction
Let us start with the following inequality for the Legendre polynomials P n ( x ) − P n − ( x ) P n +1 ( x ) ≥ , where x ∈ [ − ,
1] and n ∈ { , . . . } , which was first proved by the Hungarian mathematician Paul Tur´an[15] while studying the zeros of Legendre polynomials. Thereafter an inequality of this kind is known asa Tur´an type inequality. G. Szeg˝o [14] has given four elegant proofs of the above inequality. There isan immense literature on the Tur´an type inequalities for classical (orthogonal) polynomials and specialfunctions. For recent development on the topic we suggest the reader to refer [2]-[12] and referencestherein.In this paper we study the Tur´an type inequalities for the confluent hypergeometric functions of thesecond kind denoted by ψ ( a, c, · ) also known as Tricomi confluent hypergeometric functions. This functionis a particular solution of Kummer’s differential equation [1, p. 504] xy ′′ ( x ) + ( c − x ) y ′ ( x ) − ay ( x ) = 0 , and for a > c ∈ R , and x > ψ ( a, c, x ) = 1Γ( a ) Z ∞ e − xt t a − (1 + t ) c − a − dt. Recently Baricz and Ismail [8] have deduced some Tur´an type inequalities for confluent hypergeometricfunctions of the second kind. In [7], the author found some tight bounds for Tur´anian of modified Besselfunctions of first and second kind. Motivated by the results from [7], in this paper we find tight boundsfor the Tur´anians of confluent hypergeometric functions of the second kind and we offer some alternativeproofs of the Tur´an type inequalities given in [8]. Moreover, by using a technique similar to [3], we derivesome new inequalities for confluent hypergeometric functions of the second kind. The organization of thepaper is as follows: in Sect. 2 we state our main results, while in Sect. 3 we prove the main results andwe give some alternative proofs of the Tur´an type inequalities derived in [8].2.
Main Results
Our first set of results deals with some tight Tur´an type inequalities for confluent hypergeometricfunctions of the second kind, which improve the existing Tur´an type inequalities proved in [8]. The firstmain result is related to the bounds of the Tur´anian of confluent hypergeometric functions of the secondkind when we shift both parameters.
File: Tricomi.tex, printed: 2018-07-24, 12.57
Mathematics Subject Classification.
Key words and phrases.
Confluent hypergeometric functions of the second kind, Tur´an type inequalities. ⋆ The work of ´A. Baricz was supported by the J´anos Bolyai Research Scholarship of the Hungarian Academy of Sciences.The second author is on leave from the Department of Mathematics, Indian Institute of Technology Madras, Chennai-600036, India. The research of S. Singh was supported by the fellowship of the University Grants Commission, India.
Theorem 1. If a > , c < and x > , then the following Tur´an type inequality is valid (2.1) c − a − x ψ ( a, c, x ) < ψ ( a, c, x ) − ψ ( a − , c − , x ) ψ ( a + 1 , c + 1 , x ) . Similarly, if a > , c < − and x > , then we have the next inequality (2.2) ψ ( a, c, x ) − ψ ( a − , c − , x ) ψ ( a + 1 , c + 1 , x ) < (cid:18) c + 2 x (cid:18) c − ac ( c + 1) (cid:19)(cid:19) ψ ( a, c, x ) . In addition, the following Tur´an type inequality (2.3) − x ψ ( a, c, x ) < ψ ( a, c, x ) − ψ ( a − , c − , x ) ψ ( a + 1 , c + 1 , x ) holds for all a > , c < and x > . The inequalities (2.1) and (2.3) are sharp as x → ∞ , while (2.2) issharp as x → . We would like to mention here that for x > c ( c − a − c ψ ( a, c, x ) < ψ ( a, c, x ) − ψ ( a − , c − , x ) ψ ( a + 1 , c + 1 , x ) , which is valid for a > > c , and x >
0, while for x < c ( c +1)2( a − c ) , the inequality (2.2) is better than the Tur´antype inequality [8, Theorem 2](2.5) ψ ( a, c, x ) − ψ ( a − , c − , x ) ψ ( a + 1 , c + 1 , x ) < a > c < x >
0. Moreover, observe that for x > − c >
0, (2.3) is better than(2.4).The next result is about the Tur´anian of confluent hypergeometric functions of the second kind whenwe shift only the first parameter.
Theorem 2. If a > > c and x > then the following Tur´an type inequalities are valid (2.6) 11 + a − c (cid:18) c x (cid:19) ψ ( a, c, x ) < ψ ( a, c, x ) − ψ ( a − , c, x ) ψ ( a + 1 , c, x ) < x ψ ( a, c, x ) . Moreover the right-hand side of (2.6) holds true for all a > , c < and x > . In addition, the followingTur´an type inequality (2.7) 11 + a − c (cid:18) − (cid:18) c − ac ( c + 1) (cid:19) x (cid:19) ψ ( a, c, x ) < ψ ( a, c, x ) − ψ ( a − , c, x ) ψ ( a + 1 , c, x ) holds for a > , c < − and x > . All of the above inequalities are sharp as x → except right-handside of (2.6) which is sharp as x → ∞ . We also note that when x < − c , the left-hand side of (2.6) improves the Tur´an type inequality [8,Theorem 3], namely(2.8) ψ ( a, c, x ) − ψ ( a − , c, x ) ψ ( a + 1 , c, x ) > , which is valid for a > c < x >
0, while for x > a − c ), the right-hand side of (2.6) improvesthe Tur´an type inequality [8, Theorem 3](2.9) ψ ( a, c, x ) − ψ ( a − , c, x ) ψ ( a + 1 , c, x ) <
11 + a − c ψ ( a, c, x ) , which is valid a > > c and x >
0. Note also that (2.7) is better than (2.8) for a > c < − x > x < c ( c +1) c − a .Now, we focus on the Tur´anian of confluent hypergeometric functions of the second kind when we shiftonly the second parameter. Theorem 3. If a > , c < and x > then the following Tur´an type inequality is valid (2.10) − ax ψ ( a, c, x ) < ψ ( a, c, x ) − ψ ( a, c − , x ) ψ ( a, c + 1 , x ) . In addition, if a > , c < − and x > then the following Tur´an type inequality is valid (2.11) ψ ( a, c, x ) − ψ ( a, c − , x ) ψ ( a, c + 1 , x ) < ac (1 + a − c ) (cid:18) x (cid:18) c − ac ( c + 1) (cid:19)(cid:19) ψ ( a, c, x ) . The inequality (2.10) is sharp as x → ∞ , while (2.11) is sharp as x → . UR´AN TYPE INEQUALITIES FOR CONFLUENT HYPERGEOMETRIC FUNCTIONS 3
For x > c ( c − a − ac (1 + a − c ) ψ ( a, c, x ) < ψ ( a, c, x ) − ψ ( a, c − , x ) ψ ( a, c + 1 , x ) , which is valid for all a > > c and x >
0, while for x < c ( c +1)2( a − c ) , the inequality (2.11) is better than theTur´an type inequality [8, Theorem 4] ψ ( a, c, x ) − ψ ( a, c − , x ) ψ ( a, c + 1 , x ) < , which is valid for a, x > c ∈ R .It is worth also to mention that very recently Simon [13] obtained also some interesting Tur´an typeinequalities for Tricomi confluent hypergeometric functions, however these inequalities cannot be directlycomparable with our present results. In [13] the following Tur´an type inequalities appear ψ ( a, c, x ) − ψ ( a, c − , x ) ψ ( a, c + 1 , x ) ≥ − x ψ ( a, c, x ) ψ ( a, c − , x ) ,ψ ( a, c, x ) − ψ ( a, c − , x ) ψ ( a, c + 1 , x ) ≥ − x ψ ( a, c, x ) ψ ( a + 1 , c + 1 , x ) , where a > , c < a + 2 , x > a > , c < a + 1 , x > Theorem 4. If a > > c and x > , then the following inequalities are valid: (2.13) (cid:18) Γ( a − c + 1)Γ( − c ) ψ ( a + 1 , c + 1 , x ) (cid:19) a +1 < (cid:18) Γ( a − c + 1)Γ(1 − c ) ψ ( a, c, x ) (cid:19) a , and (2.14) 2 < ψ ( a, c, x ) ψ ( a + 1 , c + 1 , x ) − c (cid:18) Γ( a − c + 1)Γ(1 − c ) ψ ( a, c, x ) (cid:19) a . Moreover, the following inequality (2.15) (cid:18) Γ( a − c + 1)Γ(1 − c ) ψ ( a, c, x ) (cid:19) ca ( c +1) < (cid:18) Γ( a − c + 1)Γ( − c ) ψ ( a + 1 , c + 1 , x ) (cid:19) a +1 is valid for a > , c < − , x > and the inequality (2.16) ψ ( a + 1 , c + 1 , x ) < − c ψ ( a, c, x ) is valid for a > > c and x > . Proofs of main results
In this section we prove our main results and we also give alternative proofs for the Tur´an typeinequalities (3 . . .
15) and left-hand side of (3 .
16) in [8].
Proof of Theorem 1.
Taking into account the proof of [8, Theorem 2], we know that for a > c < x > ψ ∆ a,c ( x ) ψ ( a, c, x ) = − Z ∞ tϕ a,c ( t )( x + t ) dt, where ϕ a,c ( t ) = t − c e − t | ψ ( a, c, te iπ ) | − Γ( a + 1)Γ( a − c + 1) , and ψ ∆ a,c ( x ) = ψ ( a, c, x ) − ψ ( a − , c − , x ) ψ ( a + 1 , c + 1 , x ) . Observe that (cid:18) x ψ ∆ a,c ( x ) ψ ( a, c, x ) (cid:19) ′ = − Z ∞ xt ϕ a,c ( t )( x + t ) dt < , for all a > c < x >
0. Thus, the function x x ψ ∆ a,c ( x ) ψ ( a, c, x ) ´A. BARICZ, S. PONNUSAMY, AND S. SINGH is strictly decreasing and maps (0 , ∞ ) into ( −∞ , . Consequently we have for all a > c < x > ζ a,c := lim x →∞ x ψ ∆ a,c ( x ) ψ ( a, c, x ) < x ψ ∆ a,c ( x ) ψ ( a, c, x ) < lim x →∞ x ψ ∆ a,c ( x ) ψ ( a, c, x ) =: η a,c . Now to compute ζ a,c , we use the asymptotic expansion [1, p. 508] ψ ( a, c, x ) ∼ x − a (cid:18) a ( c − a −
1) 1 x + 12 a ( a + 1)( a + 1 − c )( a + 2 − c ) 1 x + · · · (cid:19) , which is valid for large real x and fixed a and c . Thus, we have x ψ ∆ a,c ( x ) ψ ( a, c, x ) ∼ x (cid:0) α ( a, c ) x + α ( a, c ) x + · · · (cid:1) · (cid:18) α ( a, c ) 1 x + α ( a, c ) 1 x + · · · (cid:19) − (cid:18) α ( a − , c −
1) 1 x + α ( a − , c −
1) 1 x + · · · (cid:19) × (cid:18) α ( a + 1 , c + 1) 1 x + α ( a + 1 , c + 1) 1 x + · · · (cid:19)(cid:19) , where α ( a, c ) = a ( c − a −
1) and α ( a, c ) = 12 a ( a + 1)( a + 1 − c )( a + 2 − c ) . Since 2 α ( a, c ) = α ( a − , c −
1) + α ( a + 1 , c + 1), it follows that in the above asymptotic expansioninside of square brackets the factor 1 /x vanishes and thus we just need to look at the coefficient of thefactor 1 /x . But this is2 α ( a, c ) + α ( a, c ) − α ( a − , c − − α ( a + 1 , c + 1) − α ( a − , c − α ( a + 1 , c + 1) . After some algebra we obtain that the above expression becomes c − a −
1, and thus we have that(3.2) ζ a,c = lim x →∞ x ψ ∆ a,c ( x ) ψ ( a, c, x ) = c − a − , that is, indeed the inequality (2.1) is valid.Note that in view of the proof of [8, Theorem 2] we know that(3.3) lim x → ψ ∆ a,c ( x ) ψ ( a, c, x ) = 1 c for a > > c , and thus it follows that η a,c = lim x → x . ψ ∆ a,c ( x ) ψ ( a, c, x ) = 0 . In this case we obtain the Tur´an type inequality (2.5).Alternatively, (2.1) can be proved as follows. Using (3.2) and (3.1) we have(3.4) Z ∞ tϕ a,c ( t ) = 1 + a − c. Now from (3.1) and (3.4), for all a > c < x > ψ ∆ a,c ( x ) ψ ( a, c, x ) = − Z ∞ tϕ a,c ( t )( x + t ) dt > − Z ∞ tϕ a,c ( t ) x = c − a − x . This gives another proof of (2.1).Moreover, again using (3.1), (3.4) and the inequality ( x + t ) > x , we can get another proof of (2.1).Namely, we have that (cid:18) ψ ∆ a,c ( x ) ψ ( a, c, x ) (cid:19) ′ = Z ∞ tϕ a,c ( t )( x + t ) dt < Z ∞ tϕ a,c ( t ) x dt = (cid:18) c − a − x (cid:19) ′ , that is, the function x ψ ∆ a,c ( x ) ψ ( a, c, x ) − c − a − x is strictly decreasing on (0 , ∞ ). Since (see [8])(3.5) lim x →∞ ψ ∆ a,c ( x ) ψ ( a, c, x ) = 0 , UR´AN TYPE INEQUALITIES FOR CONFLUENT HYPERGEOMETRIC FUNCTIONS 5 it results that ψ ∆ a,c ( x ) ψ ( a, c, x ) > c − a − x , which is exactly (2.1).By using a similar trick as above we have that(3.6) Z ∞ ϕ a,c ( t ) = 1 , since (see proof of [8, Theorem 3]) lim x →∞ ψ ∆ a ( x ) ψ ( a, c, x ) = 0 , and (1 + a − c ) ψ ∆ a ( x ) ψ ( a, c, x ) = 1 − Z ∞ x ϕ a,c ( t )( x + t ) dt, where ψ ∆ a ( x ) = ψ ( a, c, x ) − ψ ( a − , c, x ) ψ ( a + 1 , c, x ), a > c < x > tx < ( t + x ) we obtain ψ ∆ a,c ( x ) ψ ( a, c, x ) = − Z ∞ tϕ a,c ( t )( x + t ) dt > − x Z ∞ ϕ a,c ( t ) dt = − x ;or equivalently ψ ( a, c, x ) − ψ ( a − , c − , x ) ψ ( a + 1 , c + 1 , x ) > − x ψ ( a, c, x ) , where a > c < x >
0. This proves the Tur´an type inequality (2.3).Taking into account of the differentiation formula [1, p. 507](3.7) ψ ′ a,c,x ( x ) = − aψ ( a + 1 , c + 1 , x ) , we obtain (cid:18) ψ ∆ a,c ( x ) ψ ( a, c, x ) (cid:19) ′ = ( a − ψ ( a, c, x ) ψ ( a + 1 , c + 1 , x ) + ( a + 1) ψ ( a − , c − , x ) ψ ( a + 2 , c + 2 , x ) ψ ( a, c, x ) ψ ( a, c, x ) − aψ ( a − , c − , x ) ψ ( a + 1 , c + 1 , x ) ψ ( a, c, x )and in view of the asymptotic expansion [1, p. 508](3.8) ψ ( a, c, x ) ∼ Γ(1 − c )Γ( a − c + 1) , where a > c < x →
0, we obtain that (cid:18) ψ ∆ a,c ( x ) ψ ( a, c, x ) (cid:19) ′ ∼ ( a − (1 − c )Γ( − c ) + ( a + 1)Γ(1 − c )Γ(2 − c )Γ( − − c ) − a Γ(2 − c )Γ ( − c )Γ (1 − c ) , where a > c < − x →
0. After some algebra (using Γ( α + 1) = α Γ( α )) we get (cid:20) ψ ∆ a,c ( x ) ψ ( a, c, x ) (cid:21) ′ ∼ c − a ) c ( c + 1) , where a > c < −
1. On the other hand, using (3.1) we obtainlim x → (cid:18) ψ ∆ a,c ( x ) ψ ( a, c, x ) (cid:19) ′ = 2 Z ∞ ϕ a,c ( t ) t dt. Thus we have(3.9) Z ∞ ϕ a,c ( t ) t dt = c − ac ( c + 1) , where a > c < −
1. Now in view of (3.9) and the inequality ( x + t ) > t we have (cid:18) ψ ∆ a,c ( x ) ψ ( a, c, x ) (cid:19) ′ = Z ∞ tϕ a,c ( t )( x + t ) dt < Z ∞ ϕ a,c ( t ) t dt = (cid:18) c − a ) c ( c + 1) x (cid:19) ′ , ´A. BARICZ, S. PONNUSAMY, AND S. SINGH and thus the function x ψ ∆ a,c ( x ) ψ ( a, c, x ) − x (cid:18) c − ac ( c + 1) (cid:19) is strictly decreasing on (0 , ∞ ) for a > c < −
1. Consequently in view of (3.3) the Tur´an typeinequality (2.2) follows.Finally, observe that in view of (3.3) the inequality (2.2) is sharp as x → , while (3.5) gives thesharpness of (2.1) and (2.3) as x → ∞ . (cid:3) Proof of Theorem 2.
Let us consider the Tur´anian ψ ∆ a ( x ) = ψ ( a, c, x ) − ψ ( a − , c, x ) ψ ( a + 1 , c, x ) . According to the proof of [8, Theorem 3] we know that(3.10) (1 + a − c ) ψ ∆ a ( x ) ψ ( a, c, x ) = 1 − Z ∞ x ϕ a,c ( t )( x + t ) dt, where a > c < x > x x (cid:16) (1+ a − c ) ψ ∆ a ( x ) ψ ( a,c,x ) − (cid:17) . Since for all x > t > (cid:18) x ( x + t ) (cid:19) ′ = 2 x ( x + 2 t )( x + t ) > , and thus the above function is strictly increasing on (0 , ∞ ). Since (see the proof of [8, Theorem 3])lim x → (1 + a − c ) ψ ∆ a ( x ) ψ ( a, c, x ) = 1 , for a > > c and x >
0, it follows thatlim x → x (cid:18) (1 + a − c ) ψ ∆ a ( x ) ψ ( a, c, x ) − (cid:19) = 0 , and thus we have the inequality (2.9), which is exactly the left-hand side of (3.15) in [8].By using (3.4), (3.10) and the inequality x ( x + t ) < x we have (cid:18) ψ ∆ a ( x ) ψ ( a, c, x ) (cid:19) ′ = −
11 + a − c Z ∞ xtϕ a,c ( t )( x + t ) dt > −
11 + a − c Z ∞ tϕ a,c ( t ) x dt = (cid:18) x (cid:19) ′ , that is, the function x ψ ∆ a ( x ) ψ ( a, c, x ) − x is strictly increasing on (0 , ∞ ). Since (see the proof of [8, Theorem 3])lim x →∞ ψ ∆ a ( x ) ψ ( a, c, x ) = 0 , we deduce that ψ ( a, c, x ) − ψ ( a − , c, x ) ψ ( a + 1 , c, x ) < x ψ ( a, c, x ) , where a > c < x >
0. This proves the right-hand side of (2.6).We also note that (3.6) can be used to get another proof of (2.8), which is the right-hand side of (3.15)in [8]. Namely, for a > c < x > a − c ) ψ ∆ a ( x ) ψ ( a, c, x ) = 1 − Z ∞ x ϕ a,c ( t )( x + t ) dt > − Z ∞ ϕ a,c ( t ) = 0 , which gives right-hand side of (3 .
15) in [8].Now, observe that in view of (3.1) and (3.3) we have(3.11) lim x → ψ ∆ a,c ( x ) ψ ( a, c, x ) = 1 c = − Z ∞ ϕ a,c ( t ) t dt, where a > > c . From (3.11) and the inequality ( x + t ) > xt we have for all a > > c and x > a − c ) ψ ∆ a ( x ) ψ ( a, c, x ) = 1 − Z ∞ x ϕ a,c ( t )( x + t ) dt > − x Z ∞ ϕ a,c ( t ) t dt = 1 + x c , which proves the left-hand side of (2.6). UR´AN TYPE INEQUALITIES FOR CONFLUENT HYPERGEOMETRIC FUNCTIONS 7
Now using (3.10), inequality ( x + t ) > t and (3.9), we have for all a > c < − (cid:18) (1 + a − c ) ψ ∆ a ( x ) ψ ( a, c, x ) (cid:19) ′ = − Z ∞ xtϕ a,c ( t )( x + t ) dt > − x Z ∞ ϕ a,c ( t ) t dt = (cid:18) − (cid:18) c − ac ( c + 1) (cid:19) x (cid:19) ′ . Thus, the function x (1 + a − c ) ψ ∆ a ( x ) ψ ( a, c, x ) + (cid:18) c − ac ( c + 1) (cid:19) x is strictly increasing on (0 , ∞ ) and hence using the limitlim x → (1 + a − c ) ψ ∆ a ( x ) ψ ( a, c, x ) = 1 , the Tur´an type inequality (2.7) follows.Since (see proof of [8, Theorem 3]) lim x → ψ ∆ a ( x ) ψ ( a, c, x ) = 11 + a − c and lim x → ψ ∆ a ( x ) ψ ( a, c, x ) = 0 , the left-hand side of (2.6), and the inequality (2.7) are clearly sharp as x → , while the right-hand sideof (2.6) is sharp as x → ∞ . (cid:3) Proof of Theorem 3.
By using (3.4), the inequality ( x + t ) > x and the next relation (see the proofof [8, Theorem 4]) (cid:18) ψ ∆ c ( x ) ψ ( a, c, x ) (cid:19) ′ = a a − c Z ∞ tϕ a,c ( t )( x + t ) dt, where ψ ∆ c ( x ) = ψ ( a, c, x ) − ψ ( a, c − , x ) ψ ( a, c + 1 , x ) , we get for all a > c < x > (cid:18) ψ ∆ c ( x ) ψ ( a, c, x ) (cid:19) ′ < (cid:16) − ax (cid:17) ′ , that is, the function x ψ ∆ c ( x ) ψ ( a, c, x ) + ax is strictly decreasing on (0 , ∞ ). Since (see the proof of [8, Theorem 4])(3.12) lim x →∞ ψ ∆ c ( x ) ψ ( a, c, x ) = 0and hence we get the inequality (2.10).By using (3.11) and the inequality ( x + t ) > t we obtain ψ ∆ c ( x ) ψ ( a, c, x ) = − a a − c Z ∞ tϕ a,c ( t )( x + t ) dt > − a a − c Z ∞ ϕ a,c ( t ) t dt = ac (1 + a − c ) , where a > > c and x >
0. This is the left-hand side of (3.16) in [8], that is, the inequality (2.12).Finally, using (3.9) for a > c < − x > (cid:18) ψ ∆ c ( x ) ψ ( a, c, x ) (cid:19) ′ = a a − c Z ∞ tϕ a,c ( t )( x + t ) dt < a a − c Z ∞ ϕ a,c ( t ) t dt = (cid:18) a a − c (cid:18) c − ac ( c + 1) (cid:19) x (cid:19) ′ , that is, the function x ψ ∆ c ( x ) ψ ( a, c, x ) − a a − c (cid:18) c − ac ( c + 1) (cid:19) x is decreasing on (0 , ∞ ). Since (see the proof of [8, Theorem 4])(3.13) lim x → ψ ∆ c ( x ) ψ ( a, c, x ) = ac (1 + a − c ) , we get the inequality (2.11).Now in view of (3.12), the inequality (2.10) is sharp as x → ∞ while (3.13) gives the sharpness of(2.11) as x → (cid:3) ´A. BARICZ, S. PONNUSAMY, AND S. SINGH Proof of Theorem 4.
To prove (2.13), consider the function f a,c : (0 , ∞ ) → R , defined by f a,c ( x ) = 1 a log ψ ( a, c, x ) − a + 1 log ψ ( a + 1 , c + 1 , x ) , which in view of the Tur´an type inequality (2.5) and the differentiation formula (3.7) gives f ′ a,c ( x ) = ψ ( a + 2 , c + 2 , x ) ψ ( a + 1 , c + 1 , x ) − ψ ( a + 1 , c + 1 , x ) ψ ( a, c, x ) > . Therefore f a,c is increasing on (0 , ∞ ) and hence f a,c ( x ) > f a,c (0) for x > ψ ( a + 1 , c + 1 , x ) ψ ( a, c, x ) < (cid:18) Γ( − c )Γ( a − c + 1) (cid:19) a +1 (cid:18) Γ( a − c + 1)Γ(1 − c ) (cid:19) a ( ψ ( a, c, x )) − aa ( ψ ( a + 1 , c + 1 , x )) aa + a < − c (cid:18) Γ( a − c + 1)Γ(1 − c ) ψ ( a, c, x ) (cid:19) a , which by using the following arithmetic-geometric mean inequality2 ≤ ψ ( a, c, x ) ψ ( a + 1 , c + 1 , x ) + ψ ( a + 1 , c + 1 , x ) ψ ( a, c, x )gives the inequality (2.14).Now, to prove the inequality (2.15), consider the function g a,c : (0 , ∞ ) → R , defined by g a,c ( x ) = ca ( c + 1) log ψ ( a, c, x ) − a + 1) log ψ ( a + 1 , c + 1 , x ) , which in view of the Tur´an type inequality (2.4) and the differentiation formula (3.7) gives g ′ a,c ( x ) = ψ ( a + 2 , c + 2 , x ) ψ ( a + 1 , c + 1 , x ) − cc + 1 ψ ( a + 1 , c + 1 , x ) ψ ( a, c, x ) < . Therefore g a,c is decreasing on (0 , ∞ ) and g a,c ( x ) < g a,c (0) for x > h a,c : (0 , ∞ ) → R , defined by h a,c ( x ) = log ψ ( a, c, x ) − log ψ ( a + 1 , c + 1 , x ) , which according to the following Tur´an type inequality for confluent hypergeometric functions of thesecond kind [8](3.14) ψ ( a, c, x ) − ψ ( a − , c − , x ) ψ ( a + 1 , c + 1 , x ) < a ψ ( a, c, x ) , valid for a > c ∈ R and x >
0, and the differentiation formula (3.7) gives h ′ a,c ( x ) = ψ ( a + 2 , c + 2 , x ) ψ ( a + 1 , c + 1 , x ) − (cid:18) aa + 1 (cid:19) ψ ( a + 1 , c + 1 , x ) ψ ( a, c, x ) > . Hence h a,c is increasing on (0 , ∞ ) and consequently in view of (3.8) inequality (2.16) follows.We note that the Tur´an type inequality (3.14) has been proved in [8] using the classical H¨older-Rogers inequality for integrals. Alternatively, this inequality can be proved using convolution theoremfor Laplace transforms. Namely, by taking into account of the integral representation (1.1) and theconvolution theorem for Laplace transforms, it follows that ψ ∆ a,c ( x ) = ψ ( a, c, x ) − ψ ( a − , c − , x ) ψ ( a + 1 , c + 1 , x )= 1Γ ( a ) Z ∞ e − xt (cid:18)Z t ( t − u ) a − (1 + t − u ) c − a − u a − (1 + u ) c − a − du (cid:19) dt − a − a + 1) Z ∞ e − xt (cid:18)Z t ( t − u ) a − (1 + t − u ) c − a − u a (1 + u ) c − a − du (cid:19) dt = 1Γ ( a ) Z ∞ e − xt (cid:18)Z t ( t − u ) a − (1 + t − u ) c − a − u a − (1 + u ) c − a − (cid:18) t − u a − a (cid:19) du (cid:19) dt. UR´AN TYPE INEQUALITIES FOR CONFLUENT HYPERGEOMETRIC FUNCTIONS 9
Now by using the change of variable u = t (1+ y )2 , it follows that, Z t ( t − u ) a − (1 + t − u ) c − a − u a − (1 + u ) c − a − (cid:18) t − u a − a (cid:19) du = Z − (cid:18) t (1 − y )2 (cid:19) a − (cid:18) t (1 − y )2 (cid:19) c − a − (cid:18) t (1 + y )2 (cid:19) a − (cid:18) t (1 + y )2 (cid:19) c − a − y (1 − a ) + 12 a t dy = 2 Z t a − a − (cid:0) − y (cid:1) a − (cid:18) t (cid:19) − t y )4 ! c − a − a (cid:0) − a ) y (cid:1) dy. On the other hand, by using similar steps, ψ ( a, c, x ) /a can be rewritten as1 a Γ ( a ) Z ∞ e − xt (cid:18)Z t ( t − u ) a − (1 + t − u ) c − a − u a − (1 + u ) c − a − du (cid:19) dt = 1 a Γ ( a ) Z ∞ e − xt Z − (cid:18) t (1 − y )(1 + y )4 (cid:19) a − (cid:18) t (1 − y )2 (cid:19) c − a − (cid:18) t (1 + y )2 (cid:19) c − a − t dydt = 1 a Γ ( a ) Z ∞ e − xt Z t a − a − (cid:0) − y (cid:1) a − (cid:18) t (cid:19) − t y )4 ! c − a − a (cid:0) − y (cid:1) dy dt. Since for all a > y ∈ (0 , a (cid:0) − a ) y (cid:1) < a (1 − y ) , it follows that ψ ∆ a,c ( x ) < a ψ ( a, c, x ), that is, we have the Tur´an type inequality (3.14) for a > c ∈ R and x > , which is exactly the inequality (3.11) in [8]. (cid:3) References [1]
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Department of Economics, Babes¸-Bolyai University, Cluj-Napoca 400591, RomaniaInstitute of Applied Mathematics, John von Neumann Faculty of Informatics, ´Obuda University, 1034 Bu-dapest, Hungary
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