#### 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 –
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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 ().
Bσ (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
Bσ (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
≤ Bσ (x + , x + ) – Bσ (x + , x + )
× Bσ (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) – Bσ (x + x, y)
× Bσ (x, y)Bσ (x, y + y) – Bσ (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) – Bσ (α + m, β + m)
× Bσ (α, β)Bσ (α + p, β + q) – Bσ (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.
Acknowledgements
The work was supported by the Deanship of Scientific Research, King Faisal University, Saudi Arabia, through the project
no. 150244.
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