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.

Mars - Mars represents our will power, courage, ability to take actions, aggressive nature, anger, our fighting ability, brother, male friends, a boy friend for a girl, a soldier, an athlete and real estate etc.

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

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, Mars & Mercury 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-Mars-Mercury Conjunction - Prima-facie this conjunction shows someone who has a fiery speech. Mercury which represents communication is conjunct with 2 hot planets Sun & Mars. It makes person to speak in a very agitated way. When they are angry, they can speak without much thinking. It shows heated arguments. Even when they write, their way of writing is very aggressive and they end-up writing some thought-provoking things. If Mercury is in between Sun-Mars degree-wise then it will create kind of Paap Kartari Yoga for Mercury as gentle Mercury will be in between 2 cruel/malefic planets. In this situation, person's communication ability gets seriously damaged in early life. He can have diseases related with speech disorder or nervous system. In such cases, communications improve in 30s. Sun represents Career and Father. Mars is Soldier and Mercury is Communication & Travelling. So, their own Career or Father's Career may be about fighting, either through services like Army, Police, Security or through communications, like being Advocate or Activist. They may travel a lot due to their work. Actually, at different times, they can have 2 completely different careers in either fields I suggested. It may not be a good position for siblings as Mercury is with enemy Mars and lots of arguments can take place. It is a good placement for any career where valor and courage is needed in extreme as Mars's courage is further enhanced by Sun's light. 

Who will win the Conjunction? - Mercury is neutral planet, so it depends on the dignity of other two. Whichever has the best dignity, will rule over the conjunction. 

Dignity & Strength - 

This conjunction can work best in Aries, Leo, Sagittarius and Pisces as Sun-Mars both are in good dignity here and Mercury will take up the characteristics of highly dignified Sun-Mars. 

In rest of the signs, at least 1 or 2 planets will be in weak dignity and result would depend on the planets with better dignity and dashas.