Exponential inequalities for positive linear mappings

In this article, we present exponential-type inequalities for positive linear mappings and Hilbert space operators, by means of convexity and the Mond-Pe\v cari\'c method. The obtained results refine and generalize some known results. As an application, we present extensions for operator-like geometric and harmonic means.

In Propsoition 2.2, we present a special case (1.2) for a particular choice of α, however we present a simple proof for completeness. Then as an application, we present several improve- .
which simply reads as follows where the second inequality is due to the arithmetic-geometric inequality. We refer the reader to [12] for some detailed discussion of (1.7).
Another useful observations about log-convex functions is the following. If f is log-convex Simplifying this inequality implies the following.
In this article, we present several inequalities for log-convex functions based on the Mond-Pečarić method. In particular, we present inequalities that can be viewed as exponential inequalities for log-convex functions. More precisely, we present inequalities among the quantities Another interest in this paper is to present inequalities for operator-like means when filtered through normalized positive linear maps. That is, it is known that for an operator mean σ, one has [11] ϕ(AσB) ≤ ϕ(A)σϕ(B), A, B ∈ B + (H).
In particular, we show complementary inequalities for the geometric ♯ t and harmonic ! t operatorlike means, when t < 0. Of course, when t < 0, these are not operator means. Our results can be considered as extensions of [5, Theorem 2.2].

Main Results
Now we proceed to the main results, starting with a complementary result of [9, Corollary Proof. The first and the second inequalities follow from [12, Proposition 2.1] and the fact that µ (m, M, f ) > 0. So we have to prove the other inequalities. Applying a standard functional calculus argument for the operator ϕ (A) in (1.7), we get Following [9], we have for α > 0, By setting β = 0, we obtain α = max where the generalized Kantrovich constant is defined by Proof. The result follows immediately from Proposition 2.1, be letting f (x) = x t . As another application of Proposition 2.1, we have the following bounds for operator means.
To simplify our statement, we will adopt the following notations. For a given function f : for any normalized positive linear map ψ. In particular, for the given ϕ, define Then, ψ is a normalized linear mapping and the above inequalities imply, upon conjugating with ϕ(A) 1 2 , the desired inequalities.
In particular, Corollary 2.2 can be utilized to obtain versions for the geometric and harmonic operator means, as follows.
Then, for any linear map ϕ (not necessarily normalized) and for t < 0,   Further, utilizing (1.7), we obtain the following. In this result and later in the paper, we adopt the notations: and The  Proof. We give the proof for the reader's convenience. Notice first that f being either increasing or decreasing assures that a f f ′ (t 0 ) > 0. Using a standard functional calculus in (1.6) with t = A and applying ϕ to both sides imply On the other hand, applying the functional calculus argument with t = ϕ(A) implies Noting that a f and f ′ (t 0 ) have the same sign, both desired inequalities follow from (2.4) and (2.5).
Now if f was concave, replacing f with −f and noting linearity of ϕ imply the desired inequalities for a concave function.
As an application, we present the following result, which has been shown in [7, Corollary 2.8].  implies the first inequality. The second inequality follows simlarly by letting t 0 = m+M 2 . Manipulating Proposition 2.2 implies several extensions for log-convex functions, as we shall see next.
We will adopt the following constants in Theorem A. t 1 ) and The first two inequalities of the next result should be compared with Proposition 2.1; where a reverse-type is presented now.
This proves the first two inequalities. Now, for the third inequality, assume that M − m ≥ 1 and let h 1 (t) = t 1 M −m . Then the second inequality can be viewed as Since M − m ≥ 1, it follows that h 1 is operator concave. Therefore, noting (2.8) and (1.1), we which is the desired inequality in the case M − m ≥ 1.
Now, if M − m < 1, the function h 1 is convex and monotone. Therefore, taking in account (2.8) and (2.2), we obtain which completes the proof.
For the same parameters as Theorem A, we have the following comparison too, in which the first two inequalities have been shown in Theorem A.
Proof. We prove the last inequality. Letting ψ(X) = X be a normalized positive linear map and noting that ϕ is order preserving, the fourth inequality of proposition 2.1 implies For the next result, the following constants will be used.
(by the first inequality of Proposition 2.1) which completes the proof for the case M − m ≥ 1.
(by the first inequality of Proposition 2.1) which completes the proof.
Remark 2.3. In both Theorems A and B, the constants α and α 1 can be selected to be 1, as follows. Noting that the function h in both theorems is continuous on [m, M] and differentiable on (m, M), the mean value theorem assures that a h = h ′ (t 0 ) for some t 0 ∈ (m, M). This implies α = 1, since we use the notation α ≡ α(h, t 0 ) = a h h ′ (t 0 ) . A similar argument applies for h 1 . These values of t 0 can be easily found.
Moreover, one can find t 0 so that β(h, t 0 ) = 0, providing a multiplicative version. Since this is a direct application, we leave the tedious computations to the interested reader.
Utilizing Lemma 1.1, we obtain the following exponential inequality. one can apply a functional calculus argument on (3.1). With this convention, we will use the notation The following is a refinement of of Proposition 2.2. Since the proof is similar to that of Proposition 2.2 utilizing (3.1), we do not include it here.  Notice that applying this refinement to the convex function f (t) = t −1 implies refinements of both inequalities in Corollary 2.5 as follows. Remark 3.1. The inequality (3.1) has been studied extensively in the literature, where numerous refining terms have been found. We refer the reader to [13] and [14], where a comprehensive discussion has been made therein. These refinements then can be used to obtain further refining terms for Proposition 2.2.
Further, these refinements can be applied to log-convex functions too. This refining approach leads to refinements of most inequalities presented in this article; where convexity was the key idea. We leave the detailed computations to the interested reader.

Data Availability
All data generated or analysed during this study are included in this published article. There is no experimental data in this article.