Let's 1st understand what these planets represent on their own?

Sun - Sun represents Soul, Personality, King, Government, Father, Authority, Career, Top Positions, Ego, Self Esteem, Health, Education etc.

Mercury - Mercury represents communication skills, marketing, calculative ability, younger siblings, hobbies, skills with hands, intelligence, quick decision making, logical thinking, youthfulness etc.

Rahu - It represents Illusion, Foreign Lands, Foreign Things (foreign as in unknown things. So, for someone living in Punjab, culture and rituals of Kerala is also unknown and hence Foreign), Drugs, Medicines, any type of intoxication, Fame, Wealth, Success, Obsession, Past-life incomplete Desires, Movies, Television, Online World, Cheating, Imagined Fears, Unusual Things, Unique Things, Creativity, Rule breaker. Rahu also blows things out of proportions.

3 planetary conjunction - Whenever 3 or more planets are conjunct with each other in a house/sign then things related to that house/sign become a kind of life focus for person. It is because of huge accumulation of energy within 30 degrees of sky. It means not only these 3 planets' energy is going into this house/sign but the energy of at least 4 to 6 house/sign is going into this house/sign as these planets will be ruling some houses too. So, this house/sign with conjunction becomes almost the most important house in chart. During Career Consultation, I give a huge importance to any such house which has 3 or more planets in it.

House Ruler - Besides these karaka things, Sun, Mercury & Rahu will also rule some houses in your chart and will represent things related to that house. Of course we can't cover conjunctions Ascendant and house wise else it will never be over. So, you have to include things related to the houses at the time of interpreting this conjunction. Also, we need to keep in mind all Yogas and Raj-Yogas which their conjunction would be making to completely understand the conjunction. Here, we can only cover meaning as per karakas.

Meaning of Sun-Mercury-Rahu Conjunction - This is another conjunction where person is born near solar eclipse. Closer Sun and Rahu are degree-wise, the more impact you will see of Sun being eclipsed which indicates towards loss of father, ego or confidence in early life or person can have super-inflated ego. It shows a distant or unusual relation with Father. As Mercury represents siblings, same applies for relations with siblings. Person can have very unusual or weird ways of communicating or he may prefer to communicate through online means. They may find their right career, success in career or confidence with certain delay in life which can be up to 40 years of age in some cases. They can prosper more in foreign lands or in business related with foreign lands or foreign products. They can also do some very unusual business. They can also be very good in learning foreign or machine languages. Mercury-Rahu together normally makes a person very clever and almost cunning. So, they can be very clever businessman. They can also speak a lot and big things. If Mercury is between Rahu and Sun degreewise then it creates a scenario like Paap Kartari Yoga where gentle Mercury is in between 2 strong malefic planets Rahu & Sun. In such case, person may speak without any sense and his intellect may suffer. It can also cause some illness related with nervous system. 

Who will win the Conjunction? - Rahu is impacting both the planets. So, Rahu rules over the conjunction. 

Dignity & Strength - 

Rahu's job is to explode the results of Sun-Mercury as per their dignity. So, results depend on dignity of Sun-Mercury. If Sun-Mercury are placed well in chart, Rahu will explode good side of Sun-Mercury else vice-versa.