Strongly log -biconvex Functions and Applications

In this paper, we consider some new classes of log-biconvex functions. Several properties of the log-biconvex functions are studied. We also discuss their relations with convex functions. Several interesting results characterizing the log-biconvex functions are obtained. New parallelogram laws are obtained as applications of the strongly log-biconvex functions. Optimality conditions of diﬀerentiable strongly log-biconvex are characterized by a class of bivariational inequalities. Results obtained in this paper can be viewed as signiﬁcant improvement of previously known results.


Introduction
Convex functions and convex sets have played an important and fundamental part in the development of various fields of pure and applied sciences. Convexity theory describes a broad spectrum of very interesting developments involving a link among various fields of mathematics, physics, economics and engineering sciences. In recent years, various extensions and generalizations of convex functions and convex sets have been considered and studied using innovative ideas and techniques. Noor and Noor [15,16,17,19] introduced Biconvex sets and biconvex functions, which inspired a great interest. They proved that the differentiable biconvex functions are biconvex functions and the converse is also true under Recently, Noor et al. [12,13,18] considered the equivalent formulation of log-convex functions and proved that the log-convex functions have similar properties as the convex functions enjoy. Inspired and motivated by the ongoing research in this interesting, applicable and dynamic field, we consider the concept of strongly log-biconvex functions. We discuss the basic properties of the log-biconvex functions. It is has been shown that the log-biconvex(biconcave) have nice properties. Several new concepts of strongly log-biconvex functions have been introduced and investigated. The difference (sum) of the strongly log-biconvex function and affine strongly log-biconvex function is again a log-biconvex function. New parallelogram laws are obtained as applications of the strongly log-biconvex affine function, which can be viewed as a novel application. We show that the local minimum of the log-convex functions is the global minimum. The optimal conditions of the differentiable strongly log-biconvex functions can be characterized by a class of variational inequalities, which is itself an interesting outcome of our main results. The ideas and techniques of this paper may be log-biconvex Functions 3 a starting point for further research in these different areas of mathematical programming, machine learning and related optimization problems.

Preliminary Results
Let K be a nonempty closed set in a real Hilbert space H. We denote by ·, · and · be the inner product and norm, respectively. Let F : K → R be a continuous function.
Polyak [21] introduced the concept of strongly convex functions in optimization and mathematical programming.

Definition 3.
A function F is said to be a strongly convex, if there exists a constant µ ≥ 0 such that Clearly every strongly convex function is a convex function, but the converse is not true. For the applications of strongly convex functions in variational inequalities, differential equations and equilibrium problems, see [6,7,8,11,12,13,14,15,16,19,20,22,24] and the references therein.
In many problems, the underlying set may not a convex set. To overcome this deficiency, Noor et al. [16,17,19] introduced the biconvex sets and biconvex functions with respect to an arbitrary bifunction, which can be viewed as important generalization of the convexity and inspired a great interest in nonlinear mathematical programming. Definition 4. [16,17,19] The set K β in H is said to be biconvex set with respect to an arbitrary bifunction β(. − .), if We would like to emphasize that, Consequently, the biconvex set K β reduces to the convex set K. Thus, K β ⊂ K. This implies that every convex set is a biconvex set, but the converse is not true.

Definition 5.
A strictly positive function F is said to be biconvex with respect to an arbitrary bifunction β(. − .), if Noor [14] has proved that u ∈ K β is a minimum of a differentiable biconvex functions F, if and only if, u ∈ K β satisfies the inequality which is known as the bivariational inequality. For the formulation, applications, numerical methods and other aspects of bivariational inequalities and related optimization problems, see [16,17,19] and the references therein. Definition 6. A strictly positive function F is said to be log-biconvex with respect to an arbitrary bifunction β(. − .) if We can rewrite the Definition 6 in the following equivalent form as Definition 7.
[13] A strictly positive function F is said to be log-biconvex with respect to an arbitrary bifunction β(. − .), if We use this equivalent (Definition 7) to discuss some new aspects of log-biconvex functions.
If log F (u) = e f (u) , then, we recover the concepts of the exponentially biconvex function, which are mainly due to Noor and Noor [9,10] as: We remark that Definition 8 can be rewritten in the following equivalent way, which is due to Antczak [2].
A function is called the exponentially bincave function f , if −f is exponentially biconvex function. For the applications and properties of exponentially biconvex functions, see [9,10,16,17,19].
We now introduce the concept of strongly log-biconvex functions and study their basic properties.
Definition 10. A strictly positive function F is said to be strongly log-biconvex with respect to an arbitrary bifunction β(. − .), if there exists a constant µ ≥ 0 such that Definition 11. A strictly positive function F on the biconvex set K β is said to be strongly log-quasi biconvex with respect to an arbitrary bifunction β(. − .), if

Definition 12.
A strictly positive function F on the biconvex set K is said to be first kind of strongly log-biconvex with respect to an arbitrary bifunction β(. − .), if From the above definitions, we have This shows that every fist kind of strongly log-biconvex function is a strongly log-biconvex function and strongly log-biconvex function is a strongly log-quasi-biconvex function. However, the converse is not true.
If t = 1, then Definitions 2, 12 and 13, we have: Condition A plays an important part in the derivation of the main results.
Definition 13. A strictly positive function F is said to be strongly affine log-biconvex function with respect to an arbitrary bifunction β( ] be an interval. We now define the log-biconvex functions on I β .
Then F is log-convex function, if and only if, One can easily show that the following are equivalent: We also need the following assumption regarding the bifunction β(· − ·), which play a crucial part in deriving our results.

Properties of log-biconvex Functions
In this section, we consider some basic properties of log-biconvex functions.
Theorem 1. Let F be a strictly log-biconvex function. Then any local minimum of F is a global minimum.
Proof. Let the log-biconvex function F have a local minimum at u ∈ K η . Assume the contrary, that is, for arbitrary small t > 0, contradicting the local minimum.
Theorem 2. If the function F on the biconvex set K β is log-biconvex, then the level set Thus epi(F ) is an biconvex set. Conversely, let epi(F ) be an biconvex set. Let u, v ∈ K β . Then (u, log F (u)) ∈ epi(F ) and (v, log F (v)) ∈ epi(F ). Since epi(F ) is a biconvex set, we must have This shows that F is a log-biconvex function.
Proof. Let u, v ∈ L α . Then u, v ∈ K β and max(log F (u), log F (v)) ≤ α. Now for t ∈ (0, 1), w = u+tβ(v−u) ∈ K β , we have to prove that u+tβ(v−u) ∈ L α . By the quasi log-biconvexity of F, we have showing that the level set L α is indeed a biconvex set.
Conversely, assume that L α is an biconvex set.
From the definition of the level set L α , it follows that Thus F is a quasi log-biconvex function. This completes the proof.
Theorem 5. Let F be a log-biconvex function. Let µ = inf u∈K F (u). Then the set E = {u ∈ K : log F (u) = µ} is an biconvex set of K β . If F is strictly log-convex, then E is a singleton.
Proof. Let u, v ∈ E. For 0 < t < 1, let w = u+tβ(v −u). Since F is a log-biconvex function, which implies that to w ∈ E. and hence E is an biconvex set. For the second part, assume to the contrary that This contradicts the fact that µ = inf u∈K F (u) and hence the result follows.
Theorem 6. If F is a log-biconvex function such that log F (v) < log F (u), ∀u, v ∈ K β , then F is a strictly quasi log-biconvex function.
Proof. By the log-convexity of the function F, ∀u, v ∈ K β , t ∈ [0, 1], we have since log F (v) < log F (u), which shows that the function F is a strictly quasi log-biconvex function.

Properties of Strongly log-biconvex Functions
In this section, we discuss some properties of the strongly log-biconvex functions.
Theorem 7. Let F be a differentiable function on the biconvex set K β and Condition C hold. Then the function F is log-biconvex function, if and only if, Proof. Let F be a strongly log-biconvex function. Then which can be written as Taking the limit in the above inequality as t → 0, we have which is (4.1), the required result. Conversely, let (4.1) hold. Then ∀u, v ∈ K β , t ∈ [0, 1], v t = u + tβ(v − u) ∈ K β and using Condition C, we have In a similar way, we have  t) and adding the resultant, we have showing that F is a strongly log-biconvex function.
Remark 2. From (4.1), we have Changing the role of u and v in the above inequality, we also have Thus, we can obtain the following inequality Theorem 7 enables us to introduce the concept of the log-bimonotone operators, which appears to be a new ones.

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Definition 16. The differential F (.) is said to be log-bimonotone, if Definition 17. The differential F (.) is said to be log-pseudo-bimonotone, if From these definitions, it follows that strongly log-bimonotonicity implies log-bimonotonicity implies log-pseudo-bimonotonicity, but the converse is not true.
Theorem 8. Let F be differentiable strongly log-biconvex function on the biconvex set K β . Let Condition C and Condition A hold. Then (4.1) holds, if and only if, Proof. Let F be a strongly log-biconvex function on the biconvex set K β . Then, from Theorem 7, we have Changing the role of u and v in (4.5), we have Adding (4.5) and (4.6), we have which shows that F is a strongly log-monotone.
Conversely, from (4.4) and Condition C, we have Using Condition C, we obtain Consider the auxiliary function from which, we have Then, from (4.8), we have Integrating (4.9) between 0 and 1, we have Thus it follows, using Condition A, that which is the required (4.1).

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We now give a necessary condition for log-pseudo biconvex function.
Theorem 9. Let F (.) be a log-pseudo bimonotone and let Condition C and Condition A hold. Then F is a log-pseudo biconvex function.
Proof. Let F (.) be a log-pseudo bimonotone. Then, Taking v = v t in (4.10) and using Condition C, we have Consider the auxiliary function which is a differentiable function. Then, using (4.11), we have Integrating the above relation between 0 to 1, we have Using Condition A, we have showing that F is a log-pseudo biconvex function.
Definition 18. The function F is said to be sharply log-pseudo biconvex, if there exists a constant µ > 0 such that Theorem 10. Let F be a sharply log-pseudo biconvex function on the biconvex set K β . Then Proof. Let F be a sharply log-pseudo biconvex function on K β . Then from which we have the required result.

Definition 19.
A function F is said to be a log-pseudo biconvex function with respect to a strictly positive bifunction B(., .), such that Theorem 11. If the function F is strongly log-biconvex function such that log F (v) < log F (u), then the function F is a strongly log-pseudo biconvex.
Proof. Since log F (v) < log F (u) and F is strongly log-convex function, then ∀u, v ∈ K η , t ∈ [0, 1], we have This shows that the function F is strongly log-biconvex function.
We now show that the difference of strongly log-biconvex function and affine strongly log-biconvex function is again a log-biconvex function.
Theorem 12. Let f be a affine strongly log-biconvex function. Then F is a strongly log-biconvex function, if and only if, g = F −f is a log-biconvex function.
Proof. Let f be an affine strongly log-biconvex function. Then From the strongly log-biconvexity of F, we have From (4.12 ) and (4.13), we have (4.14) from which it follows that This shows that g = F − f is a log-biconvex function.
The inverse implication is obvious.
We remark that, if a strictly positive function F is a strongly log-biconvex function, then we have (4.15) which is called the Wright strongly log-biconvex function.
From (4.15), we have This implies that which shows that a strictly positive function F is a multiplicative Wright strongly log-biconvex function. It is an interesting problem to study the properties and applications of the Wright log-biconvex functions.

Parallelogram Laws
In this section, we derive some new parallelogram laws for uniformly Banach spaces as a novel application of strongly log-biconvex functions.
From Definition 13, we have Taking t = 1 2 in (5.1), we have which is called the log parallelogram law for the Banach spaces involving strongly log-biconvex functions.
(I). If log F (u) = u 2 , then equation (5.2) can be written as 3) reduces to the parallelogram-like law as: which is the well known parallelogram law, which characterizes the inner product spaces.
The log-parallelogram law (5.2) characterizes the inner product spaces involving exponentially biconvex function. Also, see [6,20] for the derivation and other properties of the inner product spaces.
For suitable and appropriate choice of the function β(. − .), one can obtain a wide class of new parallelogram laws, which can be used to characterize various classes of inner products.

Optimization
We now discuss the optimality condition for the differentiable strongly log-biconvex functions, which is the main motivation of our next result.
Taking v = v t in (6.2), we have Since F is differentiable strongly log-biconvex function, so Using (6.3), we have Thus, it follows that which is the required result(6.1).

Remark 3.
We note that, if u ∈ K β satisfies the then u ∈ K β is a minimum of a strongly log-biconvex function F. The inequality of the type (6.4) is called the log-bivariational inequality and appears to be a new one. It is an interesting problem to study the existences of a solution of the log-bivariational inequalities and develop some numerical methods. For the applications, formulations, numerical methods and other aspects of variational inequalities, see [7,8,15,16,17,19,20,24] and the references therein

Conclusion
Several new classes of strongly log-biconvex functions have been introduced and their properties are investigated. It has been shown that log-biconvex functions enjoy several properties which convex functions have. Some parallelogram laws are obtained as applications of the strongly log-biconvex functions, which can be used to characterize the inner product spaces. We have shown that the minimum of the differentiable strongly log-convex functions can be characterized by a new class of variational inequalities, which is called the log-bivariational inequality. One can explore the applications of the log-bivariational inequalities in pure and applied sciences. This may stimulate further research.