On the p-Version of the Schwab-Borchardt Mean

This paper deals with a one-parameter generalization of the Schwab-Borchardt mean. The new mean is defined in terms of the inverse functions of the generalized trigonometric and generalized hyperbolic functions. The four new bivariate means are introduced as particular cases of the p-version of the Schwab-Borchardt mean. For the particular value of the parameter p, these means become either the classical logarithmic mean or the Seiffert means or the Neuman-Sándor mean. Wilkerand Huygenstype inequalities involving inverse functions of the generalized trigonometric and the generalized hyperbolic functions are also established.


Introduction
The Schwab-Borchardt mean of two numbers  ≥ 0 and  > 0, denoted by SB(, ) ≡ SB, is defined as (see [1,Thrm 8.4], [2, (2.3)]).It follows from (1) that SB(, ) is not symmetric in its arguments and is a homogeneous function of degree 1 in  and .This mean has been studied extensively in [1][2][3][4][5].The goal of this paper is to define and investigate a generalization of the Schwab-Borchardt mean SB.The new one has a form which is similar to (1) but depends on a parameter  which is used in definitions of two families of higher transcendental functions called the generalized trigonometric and the generalized hyperbolic functions.Definitions of these functions are given in Section 2. Also, we will use a notion of the -hypergeometric functions of two variables.Their definition and some basic properties are given in Section 3. Throughout the sequel, the -V of SB will be denoted by SB  .The latter is introduced in Section 4.
Therein, some basic properties of the new mean are discussed.In Section 5, we define four new bivariate means which can be considered as the generalized logarithmic mean, the generalized Seiffert means, and the generalized Neuman-Sándor mean.In the last section of this paper, we shall establish inequalities involving new means as well as Wilkerand Huygens-type inequalities involving inverse functions of the generalized trigonometric and the generalized hyperbolic functions.

Functions of Two Variables
In this section, we give the definition of the bivariate hypergeometric functions which are used in the sequel.Some results for these functions are also included here.
In what follows, the symbols R + and R > will stand for the nonnegative semiaxis and the set of positive numbers, respectively.Let  = ( 1 ,  2 ) ∈ R 2 > .By   , where we will denote the Dirichlet measure on the interval [0, 1].It is well known that   is the probability measure on its domain.Also, let  = (,) ∈ R 2 > .In [16,17], the hypergeometric function   (; ) ( ∈ R) is defined as follows: where  = (, 1 − ) and  ⋅  =  + (1 − ) are the dot product of  and .Many of the important special functions, including Gauss' hypergeometric function  and some elliptic integrals, admit the integral representation (13).
For more details, the interested reader is referred to Carlson's monograph [17].
A nice feature of the -hypergeometric function is its permutation symmetry in both parameters and variables; that is, Another remarkable property of   is its homogeneity of degree  in its variables: ( > 0).For the later use, let us also record Carlson's inequality [18, Theorem 3]: (,  ̸ = 0,  ≤ ).

Definition and Basic Properties of the 𝑝-Version of SB
Let the numbers  and  have the meaning as in Section 1.
For the sake of presentation, we recall first a formula for the mean SB in terms of the -hypergeometric function: (see [17,19]).
We define the -version ( > 1) of the mean SB as follows: The rightmost member of ( 19) is a special case of what is called in mathematical literature the -hypergeometric mean (see [2,17,18]).Using elementary properties in the hypergeometric functions, we see that SB  (, ) is the mean value of  and .Moreover, this mean is nonsymmetric and homogeneous of degree 1 in its variables.The well-known results on the -hypergeometric means lead to the conclusion that SB  is a strongly increasing function of the parameter .
For the brevity of notation, let us introduce a particular -hypergeometric function: Clearly function   is nonsymmetric and homogeneous of degree −1/ in its variables.Comparison with (19) yields We shall demonstrate now that SB  can also be expressed in terms of cos −1   and cosh −1  : For the proof of the first part of ( 22), let us record a formula which shows that the Gauss hypergeometric function  can be expressed in terms of the bivariate -hypergeometric function: (see, e.g., [17, ((5.9)-( 12))]).Application of the last formula to (7) yields where 0 <  < 1.This, in conjunction with (9), gives Letting above  = / and utilizing homogeneity of the function   , we obtain where in the last step we have utilized formula (19).This yields the first part of (22).The second part can be established in an analogous manner.A key formula needed here reads as follows: > 1.We omit further details.Function   admits an integral representation: This follows from the known result [20, (19.16.9)] where  stands for the beta function and   =  1 +  2 − .

Four New Bivariate Means Derived from SB 𝑝
The goal of this section is to define and investigate four new bivariate means.They are defined in terms of the SB  and the bivariate power mean   ( ∈ R).Recall that where ,  > 0. The power mean  0 of order 0 is usually denoted by  and is called the geometric mean.It is well known that the power mean   is a strictly increasing function of .
In the case when  = 2, these means become the classical logarithmic mean , two Seiffert means  and  (see [21,22]), and the Neuman-Sándor mean  introduced in [4].
For the later use, we introduce quantity V  , where The main result of this section reads as follows.
Proof.We begin with the proof of (36).Making use of ( 31) and ( 22), we obtain Elementary computations yield Multiplying and dividing by we obtain, using (35), We shall write the denominator of (40) using ( 10) and (43) as follows: This, in conjunction with (40) and (43), gives the desired result (36).We shall provide now a sketch of the proof of formula (39).It follows from ( 22) and (34) that Elementary computations yield Application of (10) with  =   / /2 gives cosh −1  (  / /2 ) = sinh −1  V  .This in conjunction with (45) and (46) yields the asserted result (39).The remaining two formulas for the -versions   and   of the Seiffert means can be established in an analogous manner using (32) or (33), (22), and (9).We leave it to the interested reader.The proof is complete.

Inequalities Involving the SB 𝑝 Means
This section deals with inequalities involving the SB  means.In particular, inequalities for the four means introduced in Section 5 are established.Also, we shall prove Wilker-type and Huygens-type inequalities involving inverse functions of the generalized trigonometric functions and the generalized hyperbolic functions.
Our first result reads as follows.
Theorem 2. Let the positive numbers  and  be such that  > .Then   (, ) <   (, ) . (47) Proof.We shall prove the assertion using integral formula (28) and formula (21).Let  > 1 and let  > 0. Then   > 1 and or what is the same that or what is the same that the inequality Raising both sides to the power of −1 and next applying formula (21), we obtain Letting  = / and next utilizing homogeneity of SB  , we obtain the desired result.
The four new means defined in Section 5 and the power means are comparable.We have the following.Corollary 3. Let ,   ,   ,   ,   and   be the mean values of two positive and unequal numbers.Then the following chain of inequalities is valid.
The second and the fifth inequalities can be obtained using Theorem 2 applied to two pairs of defining equations ( 31)-( 32) and ( 34)-(33).
Our next result reads as follows.
Proof.We shall obtain the desired result utilizing the following transformation for the R-hypergeometric functions [17, ((5.5)-( 19))]: where Letting  1 =  2 = 1 and  2 =  1 = 1/ and making substitutions  :=   and  :=   , we obtain Applying the permutation symmetry (see ( 14)) to the Rhypergeometric function on the left-hand side and next raising both sides of the resulting formula to the power of −1, we obtain Since −1 < −1/, Carlson's inequality ( 16) yields Combining this with (57), we obtain Utilizing ( 20) and ( 21), we can write the last inequality in the form The desired inequality (54) now follows.
Corollary 5. Let the numbers  and  be the same as in Theorem 4. Then the following inequalities hold true.
Proof.Inequality (61) can be obtained using Theorem 4 with  :=  /2 and  := .Next, we utilize formulas (31) and (32) to obtain the desired result.In a similar fashion, one can prove inequality (62) using Theorem 4 with  :=  and  :=  /2 .Making use of formulas (31) and (32) yields the assertion.The remaining inequalities (63) can be proven in an analogous manner.We omit further details.
Another inequality for the SB  mean is contained in the following.
Proof.First, we make the substitutions in ( 17) next we use (20), and finally we raise both sides of the resulting inequality to the power of −2.This gives Application of (21) gives the desired inequality (64).
Corollary 7. Assume that the positive numbers  and  are not equal.Then the following inequalities (68) hold true.
We will close this section proving Wilker-type and Huygens-type inequalities which involve inverse functions of the generalized trigonometric and hyperbolic functions.To this aim, we shall employ the following result [23].
Theorem A. Let , V, ,  be positive numbers.Assume that  and V satisfy the separation condition: Then the inequality International Journal of Mathematics and Mathematical Sciences or for some ,  ≥ 0 with + = 1.If  and V satisfy the separation condition (69) together with then inequality (70) is also valid if As in the previous sections, the letters  and  will stand for positive and unequal numbers.Also, ,  /2 and   denote the power means of  and .
For the sake of notation, we define We are in a position to prove the following.
Proof.For the proof of validity of (70) with (, V) as defined in (76), we let It follows from (53) that the separation condition (69) is satisfied.To complete the proof of (76), we utilize a wellknown fact about bivariate means.Let (, ) ≡  be a mean which is homogeneous of degree 1 in its variables.Then  (, ) =  (, )  (  ,  − ) . (83) In particular, we have )) 2/ = (cosh 2 (  2 )) This together with (36) and (37) gives the explicit formula (76) for (, V).We will show now that the first inequality in (71) is satisfied if  = 1/ and  = 1 − 1/.To this aim, we utilize (82) and write inequality (61) as follows: where  = 1/ and  = 1 − 1/.This yields  =  and  = .To obtain condition (78) of validity of (70), we substitute / = 1/( − 1) into the last inequality in (71).This completes the first part of the proof.Assume now that (, V) is the same as is defined in (79).We will prove that (70) holds true provided that condition (81) is satisfied.First, we define Again, we appeal to (53) to claim that  and V satisfy the separation condition (69).Making use of (36) and (37), we obtain an explicit formula (79) for (, V).We will show that the first inequality in (71) is satisfied if  = 1−1/ and  = 1/.To this aim, we utilize (86) and write inequality (62) as follows: where  = 1/ and  = 1 − 1/.To prove that (70) holds true if (81) is satisfied, we substitute / =  − 1 into the last inequality in (71).The assertion now follows.The remaining two cases when (, V) is defined in (77) or in (80) can be established in the analogous manner.In these cases, we have either or We leave it to the reader to complete the proof.