Some Inequalities Involving Fuzzy Complex Numbers

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1 Theory Applications o Mathematics & Computer Science Some Inequalities Involving Fuzzy Complex Numbers Sanjib Kumar Datta a,, Tanmay Biswas b, Samten Tamang a a Department o Mathematics, University o Kalyani,Kalyani, Dist-Nadia,PIN , West Bengal, India b Rajbari, Rabindrapalli, R N Tagore Road, PO- Krishnagar, Dist-Nadia, PIN , West Bengal, India Abstract In this paper we wish to establish a ew inequalities related to uzzy complex numbers which extend some stard results Keywords: Fuzzy set, uzzy number, uzzy complex number, uzzy complex conjugate number 010 MSC: 08A7, 30D35, 30D30 1 Introduction, Deinitions Notations The idea o uzzy subset µ o a set X was primarily introduced by LA Zadeh Zadeh, 1965 as a unction µ : X [0, 1] Fuzzy set theory is a useul tool to describe situations in which the data are imprecise or vague Fuzzy sets hle such situation by attributing a degree to which a certain object belongs to a set Among the various types o uzzy sets, those which are deined on the universal set o complex numbers are o particular importance They may, under certain conditions, be viewed as uzzy complex numbers A uzzy set z is deined by its membership unction µ z z which is a mapping rom the complex numbers C into [0, 1] where z is a regular complex number as z x + iy, is called a uzzy complex number i it satisies the ollowing conditions : 1 µ z z is continuous; An -cut o z which is deined as z z µ z z >, where 0 < 1, is open, bounded, connected simply connected; 3 z 1 z µ z z 1 is non-empty, compact, arcwise connected simply connected For detail on the set z as mentioned above, one may see Buckley, 1989 Using this concept o uzzy complex numbers, J J Buckley Buckley, 1989 shown that uzzy complex numbers is closed under the basic arithmetic operations In paper Buckley, 1989 we Corresponding author addresses: sanjib kr datta@yahoocoin Sanjib Kumar Datta, tanmaybiswas math@redimailcom Tanmay Biswas, samtentamang@yahooin Samten Tamang

2 S K Datta et al / Theory Applications o Mathematics & Computer Science also see the development o uzzy complex numbers by deining addition multiplication rom the extension principle which has been shown in terms o -cuts We now review some deinitions used in this paper Deinition 11 Buckley, 1989The complex conjugate z o z is deined as µ z z µ z z, where z x iy is the complex conjugate o z x + iy The complex conjugate z o a uzzy complex number z is also a uzzy complex number because the mapping z x + iy z x iy is continuous Deinition 1 Buckley, 1989The modulus z o a uzzy complex number z is deined as µ r z sup µ z z z r, where r is the modulus o z Similarly we may deine the modulus o a real uzzy number R as ollows: µ a R supµ a R a a i a > 0, a 0 i a 0 a a i a < 0 Now in the ollowing, we deine two special types o uzzy complex numbers z n nz o the uzzy complex number z, or any complex number z z n R Deinition 13 Fuzzy complex numbers z n nz are deined as µz z n µzn z In particular when n, we have µz nz µnz z It can be easily veriied that µz z µz z µz z µz z z z z z z + z but z z z z From the deinition o uzzy complex number one may easily veriy that z n nz are also uzzy complex numbers when z is a uzzy complex number It should be noted that the signiicance o Deinition 13 is completely dierent rom the deinitions o additions multiplications o uzzy complex numbers as mentioned in Buckley, 1989 In this paper we wish to establish a ew stared inequalities related to uzzy complex numbers

3 108 S K Datta et al / Theory Applications o Mathematics & Computer Science Lemmas In this section we present some lemmas which will be needed in the sequel Lemma 1 Buckley, 1989 Let z z be any two uzzy complex numbers Suppose A z z M z z respectively Then or 0 1, A S M P holds where S z z z 1, z z z P z z z 1, z z z Also z z z z are uzzy complex numbers The ollowing lemma may be deduced in the line o Lemma 1 so its proo is omitted Lemma Let z, z, z 3,, z n be any n number o uzzy complex numbers Also let A z z + z z n M z z z 3 z n respectively Then or 0 1, A S M P holds where S z z + z z n z 1, z, z 3,, z n z z z 3 z n P z z z 3 z n z 1, z, z 3,, z n z z z 3 z n Lemma 3 Buckley, 1989 I z is any uzzy complex number then z z where 0 1 z is a truncated real uzzy number Lemma 4 Kaumann & Gupta, 1985 I M N be any two real uzzy numbers then M + N M + N i M 0, N 0 then Lemma 5 Buckley, 1989 Let z number z Then where 0 1 M N M N be a uzzy complex conjugate number o a uzzy complex z z

4 S K Datta et al / Theory Applications o Mathematics & Computer Science Theorems In this section we present the main results o the paper Theorem 31 Let z z be any two uzzy complex numbers Then z z z z Proo The meaning o the inequality is that the interval z z is greater that or equal to the interval z z or 0 1 Now rom Lemma 1 Lemma 3, we get that z z z z z z z1 z z i z i, i 1, 31 Again in view o Lemma 4, we obtain rom Lemma 3 that z z z z z z z1 z z i z i, i 1, 3 Hence the result ollows rom 31 3 in view o This proves the theorem z 1 z z 1 z J J Buckley Buckley, 1989 proved the ollowing results: Theorem A Buckley, 1989 Let z z be any two uzzy complex numbers Then 1 z z z z z z z z But he Buckley, 1989 remained silent about the question when the equality holds in the inequality 1 o Theorem A In the next two theorems, we wish to generalise the results o Theorem A ind out the condition or which z z z + z holds respectively Theorem 3 Let z, z, z 3,, z n be any n number o uzzy complex numbers Then i z z + z z n z z + z z n ii z z z 3 z n z z z 3 z n Proo In view o Lemma 1, it ollows rom Theorem A that z z + z z n z z + z z n z + z + z z n z + z + z 3 + z z n z + z + z z n This proves the irst part o the theorem Similarly with the help o Lemma 1 the equality z z z z, one can easily establish the second part o the theorem

5 110 S K Datta et al / Theory Applications o Mathematics & Computer Science Remark In view o Lemma, Lemma 3 Lemma 4 it can also be shown that the intervals z z + z z n z z z 3 z n are less than or equal to the intervals z z + z z n z z z 3 z n respectively in Theorem 3 or 0 1 Theorem 33 Let z z be any two uzzy complex numbers such that z z z z then either arg z 1 arg z is an even multiple o π or z 1 z is a positive real number where z 1 z are any two members o z z respectively Proo The meaning o the equality is that the interval z z is equal to the interval z z or 0 1 Thus z z z z ie, z z z z ie, z z z z ie, z + z z + z ie, z 1 + z z 1 + z z i z i, i 1, ; which is only possible when either arg z 1 arg z is an even multiple o π or z 1 z is a positive real number Hence the theorem ollows Theorem 34 I z z are any two uzzy complex numbers with z z z z, then arg z 1 arg z dier by π or 3π where z 1 z are any two members o z z respectively Proo The meaning o the equality is that the -cuts o z z is equal to the corresponding -cuts o z z or 0 1 Now in view o Lemma 1 Lemma 3, we obtain that z z z z z + z z1 + z z i z i, i 1, 33 Similarly, z z z z z z z1 z z i z i, i 1, 34 Thereore rom it ollows that z z z z which implies that z1 + z z 1 z z i z i, i 1, which is only possible when arg z 1 arg z dier by π or 3π Thus the theorem is established Theorem 35 Let z z be any two uzzy complex numbers Then z ± z z z Proo For 0 1, we have z ± z z ± z z ± z z1 ± z z i z i, i 1, 35 We also deduce that z z z z z z z z z1 z z i z i, i 1, 36 Hence the theorem ollows rom in view o the ollowing inequality : z 1 ± z z 1 z

6 S K Datta et al / Theory Applications o Mathematics & Computer Science Theorem 36 I z z are any two uzzy complex numbers, then z z z z z z + z z Proo In order to prove this theorem, we wish to show that the interval z z is greater than or equal to the interval z z z z + z z or 0 1 From Lemma 3 Lemma 4, we get that z z z z + z z z z z z + z z z z z z + z z z + z z z z + z z + z z z 1 z z 1 z + z z z 1 + z z 1 z 1 + z z 1 z 1 + z z z i z i, i 1, 37 Since z 1 + z z 1 + z z 1 z 1 + z z, in view o Deinition 1 Deinition 13, the theorem ollows rom Theorem 37 Let z z be any two uzzy complex numbers Then z z + z z z z Proo In view o Lemma 1, Lemma 3 Lemma 4, we get or 0 1 that z z + z z z z + z z z + z + z z z + z + z z z1 + z + z1 z zi z i, i 1, z 1 + z + z 1 z z i z i, i 1, 38 Analogously we also see that z z z z z z z z z z z i z i, i 1, z 1 z z i z i, i 1, 39 1 Now in the line o Deinition 13, it ollows rom that the corresponding -cuts are equal Hence the theorem ollows as we obtain the equality o the two real uzzy numbers

7 11 S K Datta et al / Theory Applications o Mathematics & Computer Science In the next theorem we establish a ew properties o uzzy complex conjugate numbers depending on the concept o it Theorem 38 Let z be a uzzy complex conjugate number o a uzzy complex number z Then 1 z z, z ± z z ± z, 3 z z z z, 4 z z z z 5 z z Proo In view o Lemma 5 or 0 1, we obtain that z z z z or all z z Again z z µ z z > z or all z z Since z z, the irst part o the theorem ollows rom above For the second part o the theorem, we have to prove that the -cuts o z ± z are equal to the corresponding -cuts o z ± z Now it ollows rom Lemma 1 Lemma 5 that z ± z z ± z z ± z z1 ± z z i z i, i 1, z ± z z ± z z ± z z1 ± z z i z i, i 1, Thus the second part o the theorem is established in view o z 1 + z z 1 + z We also observe that z z z z z z z1 z z i z i, i 1, 310 We may also see that z z z z z z z1 z z i z i, i 1, 311 Now rom , we obtain that the corresponding -cuts are equal This proves the third part o the theorem For the ourth part o the theorem, we deduce that z z z z 1 z z 1 z 1 z z1 z i z z i, i 1, z z z z z z 1 z z 1 z z 1 z 1 z z1 z i z i, i 1, z

8 S K Datta et al / Theory Applications o Mathematics & Computer Science z Hence the -cuts o z are equal to the corresponding -cuts o z z which implies that the two uzzy complex numbers are equal Thus the ourth part o the theorem ollows Again we have rom Lemma 3 Lemma 5 that z z z or all z z z z z z or all z z Consequently the last part o the theorem ollows in view o z z 4 Open Problem As open problems, there are several scopes to investigate the theory o analyticity singularity in case o unctions o uzzy complex variables; analogously entire or meromorphic unctions o uzzy complex variables may be deined Naturally, the theory o dierent aspects o growth properties o entire meromorphic unctions, comparative growth estimates o iterated entire unctions, results related to exponent o convergence o zeros o entire unctions o uzzy complex variables etc may also be studied aresh Acknowledgement The authors are thankul to the reeree or his/her valuable suggestions towards the improvement o the paper Reerences Buckley, JJ 1989 Fuzzy complex numbers Fuzzy Sets Systems 333, Kaumann, A M Gupta 1985 Introduction to Fuzzy Arithmetic: Theory Applications Van Nostr Reinhold, New York Zadeh, LA 1965 Fuzzy sets Inormation Control 83,