Inequalities of extended beta and extended hypergeometric functions

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.


Introduction
For Re(x) > , Re(y) > , and Re(σ ) > , define the functions The function B σ is known as the extended beta function, which was introduced by Chaudhry et al. []. They discussed several properties of this extended beta functions and also established connection with the Macdonald, error, and Whittaker functions (also see []). 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 where σ ≥  and Re(c) > Re(b) > . For σ > , the series converges for all x, provided that c = , -, -, . . . . The ECHFs also have the integral representation Similarly, the extended Gaussian hypergometric functions (EGHFs) can be defined by where σ ≥ , Re(c) > Re(b) > , and |x| < . For σ > , the series converges when |x| <  and c = , -, -, . . . . The EGHFs also have the integral form 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 Hölder-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 ..

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 Consider the functions f (t) := t x-x  , g(t) := t y-y  , and .
Applying Chebyshev's integral inequality (), for the selected f , g, and p, we have which is equivalent to ().
for all real a. This will further reduce to Proof By the definition of log-convexity it is required to prove that Clearly, () is trivially true for α =  and α = . Let α ∈ (, ). It follows from () that Let p = /α and q = /(α). Clearly, p >  and p + q = pq. Thus, applying the well-known Hölder-Rogers inequality for integrals, () yields Hence the conclusion.
Again by considering p = /α  and q = /α  , by the Hölder-Rogers inequality for integrals it follows that For α  = α  = /, this inequality reduces to Let x, y >  be such that Then The Grüss inequality [], pp.-, for the integrals is given in the following lemma. Then where Our next result is the application of the Grüss inequality for the extended beta mappings.
Theorem  Let σ  , σ  , x, y > . Then Proof To prove the inequality, it is required to determine the upper and lower bounds of for t ∈ [, ] and x, y, σ  , σ  > . Clearly, f () = f () =  and g() = g() = . Now for t ∈ (, ), the logarithmic differentiation of f yields Similarly, we can show that Now setting f , g as before and h(t) =  for all t ∈ [, ] in Lemma  gives ().

Remark  Consider the functions
for t ∈ [, ], x, y, x  , y  > . Clearly, M = L =  and m = l = . Thus, from Lemma  we have the following inequality: Similarly, if f , g, and h defined as for t ∈ [, ] and α, β, m, n, p, q > , then (see []) we have M = m m n n (m + n) m+n and L = p p q q (p + q) p+q ; hence, the inequality 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.

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≥ a n x n and g(x) = n≥ b n x n , where a n ∈ R and b n >  for all n. Further, suppose that both series converge on |x| < r. If the sequence {a n /b n } 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.
Theorem  Let b ≥  and d, c > . Then following assertions for ECHFs are true: Proof From the definition of ECHFs it follows that If we denote f n = α n (c)/α n (d), then 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 → B σ (b + n, cb) is logconvex on (, ∞) and for all nonnegative integers n. From Theorem  it is clear that σ →  B σ (b + n, cb) is log-convex for c > b > , and hence (iv) is true. Let Then using the integral representation () of ECHFs, we have 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 This, together with (), implies which is equivalent to saying that the function is decreasing on (, ∞).
Remark  In particular, the decreasing property of is equivalent to the inequality Now define A logarithmic differentiation of f yields where y → ψ(y) = (y)/ (y) is the digamma function, which is increasing on (, ∞) and has the series form This implies that