Inequalities of extended beta and extended hypergeometric functions

Journal of Inequalities and Applications, Jan 2017

We study the log-convexity of the extended beta functions. As a consequence, we establish Turán-type inequalities. The monotonicity, log-convexity, log-concavity of extended hypergeometric functions are deduced by using the inequalities on extended beta functions. The particular cases of those results also give the Turán-type inequalities for extended confluent and extended Gaussian hypergeometric functions. Some reverses of Turán-type inequalities are also derived. MSC: 33B15, 33B99.

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Inequalities of extended beta and extended hypergeometric functions

Mondal Journal of Inequalities and Applications Inequalities of extended beta and extended hypergeometric functions Saiful R. Mondal We study the log-convexity of the extended beta functions. As a consequence, we establish Turán-type inequalities. The monotonicity, log-convexity, log-concavity of extended hypergeometric functions are deduced by using the inequalities on extended beta functions. The particular cases of those results also give the Turán-type inequalities for extended confluent and extended Gaussian hypergeometric functions. Some reverses of Turán-type inequalities are also derived. extended beta functions; extended hypergeometric functions; log-convexity; Turán-type inequalities 1 Introduction tx–( – t)y– exp – Chaudhry et al. []. They discussed several properties of this extended beta functions and also established connection with the Macdonald, error, and Whittaker functions (also Later, using this extended beta function, an extended confluent hypergeometric functions (ECHFs) were defined by Chaudhry et al. []. The series representation of the extended confluent hypergeometric functions is B(b, c – b) c = , –, –, . . . . The ECHFs also have the integral representation B(b, c – b)  tb–( – t)c–b– exp xt – © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Similarly, the extended Gaussian hypergometric functions (EGHFs) can be defined by ∞ Bσ (b + n, c – b) (a)n n B(b, c – b) n! x , tb–( – t)c–b–( – xt)–a exp – Note that for p = , the series () and () respectively reduce to the classical confluent hypergeometric series and the Gaussian hypergeometric series. The aim of this article is to study the log-convexity and log-convexity of the mentioned three extended functions. In particular, we give more emphasis on the Turán-type inequality [] and its reverse form. The work here is motivated by the resent works [–] in this direction and references therein. Inequalities related to beta functions and important for this study can be found in [, ]. In Section ., we state and prove several inequalities for extended beta functions. The classical Chebyshev integral inequality and the Ho¨ lder-Rogers inequality for integrals are used to obtain the main results in this section. The results in the Section . are very useful in generating inequalities for ECHFs and EGHFs, especially, the Turán-type inequality in Section .. The log-convexity and log-convexity of ECHFs and EGHFs are also given in Section .. 2 Results and discussion 2.1 Inequalities for extended beta functions In this section, applying classical integral inequalities like Chebychev’s inequality for synchronous and asynchronous mappings and the Hölder-Rogers inequality, we derive several inequalities for extended beta functions. Few inequalities are useful in the sequel to derive the Turán-type inequalities for EGHFs and ECHFs. Proof To prove the result, we need to recall the classical Chebyshev integral inequality ([], p.): If f , g : [a, b] → R are synchronous (both increase or both decrease) integrable functions and p : [a, b] → R is a positive integrable function, then p(t)f (t) dt p(t)g(t) dt ≤ p(t)f (t)g(t) dt. Inequality () is reversed if f and g are asynchronous. p(t) := tx–( – t)y– exp – Clearly, p is nonnegative on [, ]. Since (x – x)(y – y) ≥ , it follows that f (t) = (x – x)tx–x– and g (t) = (y – y)ty–y– have the same monotonicity on [, ]. Applying Chebyshev’s integral inequality (), for the selected f , g, and p, we have tx–( – t)y– exp – tx–( – t)y– exp – tx–( – t)y– exp – tx–( – t)y– exp – which is equivalent to (). Bσ (x, y) – Bσ +a(x, y)Bσ –a(x, y) ≤ , Proof By the definition of log-convexity it is required to prove that Bασ+(–α)σ (x, y) ≤ Bσ (x, y) α Bσ (x, y) –α for α ∈ [, ], σ, σ > , and fixed x, y > . Clearly, () is trivially true for α =  and α = . Let α ∈ (, ). It follows from () that   tx–( – t)y– exp – ασ + ( – α)σ t( – t) tx–( – t)y– exp – t(σ– t) dt α tx–( – t)y– exp Let p = /α and q = /( – α). Clearly, p >  and p + q = pq. Thus, applying the well-known Hölder-Rogers inequality for integrals, () yields tx–( – t)y– exp – tx–( – t)y– exp – = Bσ (x, y) α Bσ (x, y) –α. This implies that σ → Bσ (x, y) is log-convex. Choosing α = /, σ = σ – a, and σ = σ + a, inequality () gives Bσ (x, y) – Bσ +a(x, y)Bσ –a(x, y) ≤ . ≥ . Now the identity [], p., n = , , , . . . , reduces () to ≤ . Hence the conclusion.   tx–( – t)y– exp – it follows that Bσ α(x, y) + α(x, y) ≤  tx–( – t)y– exp – t(σ– t) dt ×  tx–( – t)y– exp – t(σ– t) = Bσ (x, y)α Bσ (x, y)α .  x + x , y + y Bσ   Let x, y >  be such that ma∈iRn(x + a, x – a) > . Bσ (x, y)  ≤ Bσ (x + a, y + b)Bσ (x – a, y – b) The Grüss inequality [], pp.-, for the integrals is given in the following lemma. Lemma  Let f and g be two integrable functions on [a, b]. If m ≤ f (t) ≤ M and l ≤ g(t) ≤ L for each t ∈ [a, b], where m, M, l, L are given real constants. Then D(f , g; h) := h(t)f (t)g(t) dt – h(t)f (t) dt ≤ Bσ (x + , x + ) – Bσ (x + , x + )  × Bσ (y + , y + ) – Bσ (y + , y + )  ≤ exp(–x(+σy++ σ)) . Proof To prove the inequality, it is required to determine the upper and lower bounds of f (t) := tx( – t)x exp – g(t) := ty( – t)y exp – f (t) = f (t)( – t) xtt((– –t)t+)σ . M = L = Similarly, we can show that for t ∈ [, ] and x, y, σ, σ > . Clearly, f () = f () =  and g() = g() = . Now for t ∈ (, ), the logarithmic differentiation of f yields Remark  Consider the functions f (t) = tx, g(t) = ( – t)y h(t) = tx–( – t)y– exp – for t ∈ [, ], x, y, x, y > . Clearly, M = L =  and m = l = . Thus, from Lemma  we have the following inequality:  ≤ Bσ (x, y)Bσ (x + x, y) – Bσ (x + x, y)   × Bσ (x, y)Bσ (x, y + y) – Bσ (x, y + y)  Similarly, if f , g, and h defined as f (t) := tm( – x)n, g(t) := tp( – t)q hence, the inequality  ≤ Bσ (α, β)Bσ (α + m, β + n) – Bσ (α + m, β + m)   × Bσ (α, β)Bσ (α + p, β + q) – Bσ (α + p, β + q)  mmnn ppqq · (m + n)m+n · (p + q)p+q follows from Lemma . Remark  It is evident from Theorem  and inequalities () and () that the results discussed in [, ] for classical beta functions can be replicated for the extended beta functions. 2.2 Inequalities for ECHFs and EGHFs Along with the integral inequalities mentioned in the previous section, the following result of Biernacki and Krzyż [] will be used in the sequel. Lemma  [] Consider the power series f (x) = n≥ anxn and g(x) = n≥ bnxn, where an ∈ R and bn >  for all n. Further, suppose that both series converge on |x| < r. If the sequence {an/bn}n≥ is increasing (or decreasing), then the function x → f (x)/g(x) is also increasing (or decreasing) on (, r). We note that this lemma still holds when both f and g are even or both are odd functions. B(b, c) σ (b + δ; c; x) b → B(b + δ, c) σ (b; c; x) is decreasing on (, ∞) for fixed c, x > . Proof From the definition of ECHFs it follows that where αn(t) := BσB((bb,+t n–, bt)–n!b) . fn – fn+ = ααnn((dc)) – ααnn++((dc)) = BB((bb,,dc –– bb)) BBσσ((bb ++ nn,,dc –– bb)) – BBσσ((bb ++ nn ++ ,,dc –– bb)) . Now set x := b + n, y := d – b, x := b + n + , and y := c – b in (). Since (x – x)(y – y) = c – d ≥ , it follows from Theorem  that which is equivalent to say that the sequence {fn} is increasing, and by Lemma  we can conclude that x → σ (b; c; x)/ σ (b; d; x) is increasing on (, ∞). To prove (ii), we need to recall the following identity from [], p.: Now the increasing property of x → ≥ . This, together with (), implies × tb–( – t)c–b– exp yt – tb–( – t)c–b– exp xt – tb–( – t)c–b– exp yt – tb–( – t)c–b– exp xt – where x, y ≥  and α ∈ [, ]. This proves that x → σ (b; c; x) is log-convex for x ≥ . For the case x < , the assertion follows immediately from the identity ([], p.) It is known that the infinite sum of log-convex functions is also log-convex. Thus, the log-convexity of σ → σ (b; c; x) is equivalent to showing that σ → Bσ (b + n, c – b) is logconvex on (, ∞) and for all nonnegative integers n. From Theorem  it is clear that σ → Bσ (b + n, c – b) is log-convex for c > b > , and hence (iv) is true. Let b ≥ b. Set p(t) := tb –( – t)c–b – exp(xt – t(σ–t) ), It is easy to determine that for b ≥ b, the function f is decreasing, whereas for δ ≥ , the function g is increasing. Since p is nonnegative for t ∈ [, ], by the reverse Chebyshev integral inequality () it follows that Then using the integral representation () of ECHFs, we have f (t) := and g(t) := δ . =  f (t)g(t)p(t) dt –  g(t)p(t) dt .  f (t)p(t) dt  p(t) dt p(t)f (t) dt p(t)g(t) dt ≤ p(t)f (t)g(t) dt. This, together with (), implies BB((bb,+c)δ, cσ)(b σ+(bδ;; cc;; xx)) – BB((bb ,+c)δ, cσ)(b σ+(bδ;; cc;; xx)) ≥ , which is equivalent to saying that the function B(b, c) σ (b + δ; c; x) b → B(b + δ, c) σ (b; c; x) is decreasing on (, ∞). Remark  In particular, the decreasing property of B(b, c) σ (b + δ; c; x) b → B(b + δ, c) σ (b; c; x) is equivalent to the inequality A logarithmic differentiation of f yields where y → ψ (y) = has the series form This implies that Thus, f is a decreasing function of δ on [, ∞), and f (δ) ≤ f () = . Interestingly, for σ = , inequality () reduces to the Turán-type inequality of classical confluent hypergeometric functions we can conclude that inequality () is an improvement of the inequality given in [], Theorem (b), for fixed c, x > . However, our result does not expound the other cases in [], Theorem (b). Now following the remark given in [], p., for integer δ and b = δ + a in (), will also improve inequality ([], Theorem , Corollary ), for classical confluent hypergeometric functions. Our next result is on the extended Gaussian hypergeometric functions (EGHFs). Proof Cases (i)-(iii) can be proved by following the proof of Theorem  and considering the series form () and an integral representation () of EGHFs, we omit the details. From a result of Karp and Sitnik [] we know that if f (a, x) = n≥ m≥ has negative power series coefficient φm < , so that a → f (a, x) is strictly log-convex for x >  if the sequence {fn/fn–} is increasing. In what follows, we use this result for the function Fσ (a, b; c; x). For this, let fn = dn – dn– = BσB(σb(+b +n n–,c,–c –b)b) – BBσσ ((bb ++ nn –– ,, cc –– bb)) . Now if we replace x, y, x, y in () by x = b + n, x = b + n – , and y = y = c – b, then it follows that dn ≥ dn–. Hence the conclusion. 3 Conclusion In this article, we prove several properties of the extended beta functions resembling the classical beta functions. A few of those properties are a key to establish inequalities for ECHFs and EGHFs. Using classical integral inequalities, we also give Turán-type and reverse Turán-type inequalities for ECHFs and EGHFs. List of abbreviations ECHFs: Extended confluent hypergeometric functions; EGHFs: Extended Gaussian hypergeometric functions. Competing interests The author declares that he has no competing interests. 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Saiful Mondal. Inequalities of extended beta and extended hypergeometric functions, Journal of Inequalities and Applications, 2017, 10, DOI: 10.1186/s13660-016-1276-9